Helpful NPC Thom here to tell you why the Great Old Ones hate every single Call of Cthulhu edition and want you to play a better-designed game. Simply put, the mechanics aren't good for supporting the genre. They haven't been meaningfully updated since their original publication. Call of Cthulhu 7e provides some mechanical modernization, but they don't support the genre they claim to support.
You Can't Shoot Your Way Outta This One
Every version of Call of Cthulhu has a glut of combat mechanics. I get it, I do. The game is descended from Runequest and the Basic Roleplaying line of games, and that particular cladogram has a very short branch from D&D. Ye Olden Gamers wanted D&D but more realistic, which is a yearning near to my heart. All too familiar is the desire to snatch the shoggoths and haunters of the imagination and bestow upon them edges and boundaries. Putting them to pen and dice offers a flicker of life to what otherwise remains captive by the mind. A picture is worth a thousand words; Cthulhu's 160 hit points and "21 [armor] points of trans-dimensional muck and muscle" is worth considerably less, despite its aspirations.
Fundamentally, investigators cannot meaningfully engage with creatures of the Mythos. Horrors best be left in the dark and avoided. Now, this is not entirely faithful to Lovecraft; there are certainly some monstrosities that can be driven back with appropriate firepower (often magical). The rub is not the statistics themselves--wait. For the most part, the statistics are not the issue, although I would argue that listing Cthulhu as having two attacks per round is a poor design choice. Rather, the problem lies in the prominence of the rules governing the parts of the game that should matter least. Lovecraft's work is about encountering the supernatural and the ensuing despair and hopelessness, which is decidedly undermined with a Chicago typewriter in hand and combat flowcharts.
Weighty combat rules coupled with the vestigial stealth mechanics push the game toward combat as a resolution mechanic. Speaking of which...
Roll Spot Hidden to Find the Stealth Rules
If an insane investigator fails a pushed roll the investigator may believe he or she is invisible, when in fact everyone can see them.
This amusing bit of rules aside, the interaction of investigators with their adversaries in Call of Cthulhu should be one of caution. Hiding and sneaking should take up considerably more screentime than combat. But that's not the case in Call of Cthulhu (or any traditional RPG, for that matter). In a game where combat is a last resort, one might have expected half as many rules devoted to avoiding combat. No. Flowcharts, bonus/penalty dice, actions, rounds--that complexity is relegated to the parts of the game that aren't meant to be engaged frequently. Stealth is treated to a fraction of this design space, and the rules there amount to "roll a Stealth check if the GM deems it necessary."
A single Stealth check encompasses the entirety of the mechanics. The procedurals of combat slow gameplay and take an inordinate amount of game time compared to the most vital part of Call of Cthulhu: not getting eaten by a fish! Mythos-inspired videogames like Call of Cthulhu: Dark Corners of the Earth, Call of Cthulhu, and Amnesia: The Dark Descent all place considerable weight on puzzle-solving and evading horrors. Once combat begins in the Call of Cthulhu RPG, the procedures consume considerable game time. This is understandable to some extent because combat is contentious and proper adjudication requires rules clarity. However, that care and detail is not extended to one of the most pivotal systems in the game. avoiding this contentious system!
Contrast this with the great attention provided to the chase rules in the Call of Cthulhu RPG. While simple at their core (and in some respects, poorly-written), there is considerable page space dedicated to adjudicating chase scenes and methods for making them interesting.
You Can Run, But You Can't Hide
Unless the GM greenlights it.
Want to avoid that nasty encounter with the shoggoth and a dozen cultists? You certainly can. Maybe. How many times have you played a D&D game where the GM clearly has a setpiece battle in place and there's no possible way to avoid the clarion call of "roll for initiative"? Hopefully never, but we all know that's not the case. It's worse in Call of Cthulhu. At least (modern) editions of D&D are designed around the combat minigame, so characters are beefed up with spells, combat tricks, and hit points. Call of Cthulhu extends no such boons to the player characters, which is fitting with the genre. Most Mythos-inspired games offer the investigators a well-stocked arsenal with the assumption they're going to unloading their Chicago typewriters and sanity-draining spells into their antagonists. (Eat me, Fantasy Flight Games.) As Call of Cthulhu takes pains to dodge this design failure, one would hope they'd offer the investigators a slew of combat alternatives.
No. Eluding the Mythos' monstrosities is still entirely in the hands of the GM, regardless if you have 17% or 117% in your Stealth skill. The rules provide guidance on adjudicating these checks, but proceeding to the stage where the rules are utilized varies on the GM's perspective. Hopefully he thinks your sneaking about is possible (and doesn't feel like calling for Stealth checks every time your investigator enters a room).
Clues: Roll Perception!
Or Spot Hidden. Although Call of Cthulhu 7e takes pains to improve the lives of the investigators, the game still relies on rolling to locate clues. "Make a Perception test!" is the single most agonizing part of any traditional RPG session with an inexperienced GM. It's moreso when you're rolling to be able to progress along a mystery.
The investigators search a room and fail their Spot Hidden roll. Avoid saying, “There’s nothing there”. Tell the players they have not found anything. Now they are wondering if they missed something. Pushing the Spot Hidden roll would allow them to know if they have missed a clue or not.
This is agonizing. The rules call out the importance of making obvious clues known to players without a roll, it is enormously frustrating for a mystery to be stymied by dice mechanics. Assuming you don't take the GUMSHOE approach where the investigators automatically find clues, I would highly recommend The Alexandrian's advice on "The Art of Rulings" is to handle searching for clues. Allow me to quote a relevant section:
Sticking with our chest-searching motif, consider a scenario in which there is a hidden compartment in a chest which can be accessed by lifting out the bottom of the chest. We’ve determined that the hidden compartment requires a DC 17 Search check to discover. The player says:
“I search the chest.”
“I check the bottom of the chest for hidden compartments.”
“I take my axe and smash open the bottom of the chest.”
“I check the chest for traps before opening it.”
“I check the lid of the chest for hidden compartments”.
The first example is vanilla. You’ve handed the resolution over to the mechanic and you get a flat Search check against the DC of the hidden compartment. A success could generate a number of different responses (ranging from “yup, there’s a hidden compartment” to “you notice that the exterior of the chest is several inches deeper than the interior of the chest”).
The second example is more specific (and happens to coincide with what’s actually there to be found). I would tend to grant something like a +2 circumstance bonus to the Search. If there were other things to be found in the chest, I might also allow the Search check to find them (but since you’re specifically looking for something else, such a check would have a penalty applied to it).
In the third example you’ve taken an action which would automatically find the compartment. No Search check is required. (Player skill has completely trumped the mechanic.)
The fourth and fifth examples demonstrate that trumping character skill isn’t always a good thing. In the fourth example, you have no chance of finding things hidden inside the chest if you’re limiting your search to the exterior. Similarly, in the fifth example you’re specifically looking in the wrong place.
This alternative is still inferior to giving the players information outright, in my least humble opinion, but it provides a solid basis for adjudication.
Resolving These Problems
What kind of grumbler would I be if I dispensed only criticism and never constructive criticism? There are two primary approaches that I believe should be taken are either: less rules or more rules.
Less Is More
The less rules approach is very easy to implement, and it is oft my preference. The investigators say what they want to do, and the GM tells them how to do it, erring on the side of the affirmative. There's a cultist wandering about the house with a lantern. Investigator says, "I want to hide behind the curtains until he passes." Very well then--the investigator conceals himself behind the curtains and the cultist passes. But suppose the cultist is quite suspicious, maybe because the investigators have made a fair bit of noise in the house. Then perhaps the GM says, "You conceal yourself behind the curtains, but the cultist is wary enough that he pauses to poke about the room. How are you going to remain hidden?" The investigator has a choice on how to proceed: he can hold his breath and pray the cultist doesn't see him, relying on his skill test to save him from investigation. A risk, no doubt. Or he can act, dashing beneath the dining table when the cultist's back is turned, or he can throw something into another room to distract the cultist, or he could lunge at the cultist and try to get the upper hand.
This approach can be leveraged with increasingly stringent requirements depending on the foe. Suppose a shambling shoggoth enters the room and the player indicates he wishes to conceal himself behind the curtains. The GM says, "The shoggoth has keen senses and will surely find you, unless you [fill in the blank]." That could be unpleasantries like "spend an hour in hiding" or "toss your gun out the window so he can't sniff out the gunpowder" or "gaze upon the shoggoth and lose 1d10 sanity." Or the GM can ask the investigator what he'll do to avoid being seen and proceed from there.
Mothership RPG has no stealth mechanics whatsoever, a deliberate design choice by the author.
More Is More
The weightier alternative is to provide an in-depth conflict resolution system for these circumstances. Fate allows the GM to structure these as Conflicts, Contests, and Challenges, all with varying degrees of complexity. Rather than giving the nitty gritty on these rules mechanics (check out the SRD), even the basic four actions allow the investigators to set up a scene as follows:
The Criminal is trapped. The Mythos horror advances. It hasn't spotted him yet, but there's no hiding spot nearby. Seeing his ally in a jam, the Antiquarian studies the Mythos horror's shambling gait to sense the best moment to act. When he signals, the Athlete knocks on a table to distract the monster and dives behind cover before the monster sees him. As the monster investigates the disturbance, this in turn allows the Criminal to creep past the monstrosity.
Now, you could do this is the Call of Cthulhu game, but Fate gives each of these actions mechanical heft to make it happen with its interplay of Aspects and invocations thereof.
Another game that provides some mechanical depth for this type of Conflict is Mouseguard. While the system is abstract beyond my preference, the system involves a rock-paper-scissors style of conflict resolution wherein each "team" has a health pool that determines the duration of the conflict, and they can use various maneuvers to gain the upper hand within the conflict.
Heck, Blades in the Dark is another system. Slap a danger clock on the Mythos becoming aware of the party and check off boxes as the investigators make noise and spend time searching for clues.
Unfortunately, Call of Cthulhu opts for an uncomfortable middle ground: there are enough rules that they demand to be used, yet there are so few rules that their usage is muddled. Call of Cthulhu is begging for a mechanical modernization in its as-yet-to-be-announced eighth edition, but it is unlikely that Chaosium will provide it. Their game has been around as long as the Great Old Ones themselves, and most fans would argue that it's time-tested and doesn't need any sort of modernization.