Give GUN JAM a try. It's a rhythm shooting game, the stakes are as high or as low as you want them. There's only a minimal story and you just focus on the music, aiming and staying on the beat. Can be a lot of flow state fun.

Ravenfield with some mods on Workshop maybe? Or Operation Harshdoorstop with some Workshop.

I do have a recommendation for a cozy FPS: The Signal from Tolva.

Basically you're a robot in an old robot battlefield planet and you've got to shoot other robots, sometimes team up with other robots and go around getting upgrades.

The coziness comes from the environment, which has some strong outworldliness vibe and it feels slightly lonely but in a "journey" way. Plus the guns feel good and it's probably really cheap right now.

The cons is that the game doesn't feel finished, after a while it gets repetitive and then it just ends.

If I had to describe it, I would say it's an FPS+walking simulator.

"Generation Zero" would be my second recommendation, which is a mixture of Red Dawn, Swedish 80s and big robots.

Note: I haven't played them except for the 2nd entry, but the Far Cry series feels like it is a mixture between FPS and holiday island.

Cyberpunk on easy with aim assist is pretty chill. Popping heads and blowing off limbs. It’s a comfort game for me.

Even though they're not traditional shooters, the Ratchet and Clank games are really fun. There's a pretty big genre difference in the games in the franchise, with generally the earlier games being more platform-ey, especially 1 & 2. A lot of the humor from the original trilogy still holds up today.

There's definitely some "unfun" parts in the games (giant mech fight from rac2 and the hacker from rac3 come to mind) but overall cozy imo. Some of the most unique and fun guns in any game.

If you're looking for strictly more shooter, check out Ratchet Deadlocked. There's barely any platforming and no mini games that I can remember, it's pretty much all shooting. NG+ makes it super cozy. It's really nice to play emulated.

INCISION and ULTRAKILL

What about Extraneum? It's cute as a button and very chill with plenty of things to shoot. It feels a bit like a modern wolfenstein since there aren't any stairs. Gunplay feels nice and the secrets are rewarding. Plus between levels there are funny data discs to read.

This might be a weird answer but I find Battlefield 1 very cozy.

I felt the first Mass Effect was cozier than the sequels, though I never finished the sequels.

Combat Master, a free to play CoD like.

I've been replaying Borderlands 3, so honestly I'd recommend any of those games for a cozy shooter. The stories don't take themselves too seriously, if you die you can just respawn immediately in the vicinity, and it can be pretty relaxing to just mow down waves of bandits or whatever other enemies.

Smash Out has that aesthetic, although it can get infuriating. I also second the recommendation of Ravenfield.

Vermintide 1 is a top notch flow state game where your mind can melt while you dodge-dance and kill with fluidity. Highly recommend. Very chill.

Nothing more cozzy than shoot your friend in the head

Bulletstorm. It's not about survival as much as killing enemies as creatively as possible. And the writing is brilliant in its stupidity. One of my favourite games of all times.

payday 2 was kind of a cosy game where me and my friend would do heists whilst making tea and fetching snacks.

shadow raid? More like food raid 😊

The original Sniper Ghost Warrior games were really chill. More about stealth than action

Hedon would be an option. It has actually calm moments, is cut scene free and gives you time to explore, whil still having cool combat moments.

It although has a pretty good difficulty mode.

Maybe Destiny 2? The gunplay is unmatched and popping alien heads off in an explosion of alien blood or exploding robots in an explosion of robot blood is very satisfying.

Some parts are free to play too.

Left4Dead 2

Great flow game