No races, no beating the timer, no shooting at enemies.
Any driving or piloting games like that?
I guess the closest thing I can think of is some of the various truck simulators.
Any other suggestions?
Hopefully from games that exist on GOG.

The first thing that comes to mind is the old flight sims for Apple and Commadore. :P

Race the Sun with Sunrise DLC. Very Zen.

Arguably Big Rigs: Over the Road Racing.

You see, this game is *supposed* to be a racing game, but the enemy AI does not work at all. Your opponent will not move at all, and as a result, you *can't* lose the "race", no matter how long you take. (Winning is possible; that part of the game, where you have to go through all the checkpoints in the level, actually *does* work.) Well, they *did* release a patch that made your opponent avtually move, but it will still stop right *before* the finish line, so you *still* can't lose, even if you take your time to explore. (Also, you can't crash into your opponent (or anything else, for that matter); the game simply does not work that way.)

In any case, if this sounds like your type of game, you could look at the reviews and review scores for this game, and then decide if you want to figure out a way of acquiring the game. (The game unfortunately seems to be a bit too expensive on Amazon.)

Flight Simulator still exists (they're at X, I think with a monstrous lot of DLCs), with the same gameplay. There are also other civilian flight sims, such as X-Plane and Aerofly.

Also Simple Planes is about flying aimlessly.

Kerbal Space Program is relaxing once you get the hang of it (that music when you reach orbital peace is ultra soothing).

Abzu is about swimming but is known as a relaxing game.

And many open world games allow you to eschew combat and just go chill in the landscape.

Also : https://store.steampowered.com/app/416190/Vehicle_Simulator/

Project Zomboid, driving around the maps with a few cheats :)

Flying could be No Mans Sky in Creative mode... no enemies, no limits just you a warp drive and your imagination (and your spotify playlist of choice)

Saints Row: the Third (maybe IV) you can casually drive for hours <3 such a great little escape of a game.

in Oblivion once you get a horse you can just trot along the entire game map... drop in more world masses and if might take you an hour to cross the game world.

If you happen to own the consoles Burnout: paradise is nice, so is Mignight Club or GTAV.

Not quite piloting... but flying ala as if you have a wingsuit

https://store.steampowered.com/app/732430/Superflight/

While you get points for flying close to the mountains... there's no NEED to...

This game offers you:

Intuitive high speed flying gameplay
Endless amount of procedurally generated maps
Online Leaderboards
A seed system that lets you share maps with other players
Bright beautiful colors
Great for playing for a little while every now and then to discover some new maps and see if you can beat your old highscore

This game doesn’t have:

Helicopters
Guns
Leveling system
A Soundtrack
Lootboxes
A cinematic story
Online multiplayer
Twitch live integration
Character skins
Orcs.

I remember the original MS Flight game was quite relaxing, unfortunately, it's rather hard to get legally these days.

And I think most racing games have a mode where you're just practicing, so that would take care of the timers.

Gta3 & 5. In those you can go for a pleasant drive or fly over miles of beautiful countryside, 5 especially as you can go mountain biking in the mountains, or boating. And after a hard day relaxing you can go on a killing spree downtown to wind down.

Second this. While it's not on GOG it's completely free and open source with lots of mods being added by the community. There's cars, trucks, boats, trains, and planes (just be sure to look up the controls before playing). Some of the maps and models are incredibly detailed, but they require a decent PC to run. Of course you can turn down the settings and play with basic models as well. A lot of the trucks feature locking differentials (something I've only seen in Screamer 4x4).
If I remember correctly Screamer 4x4 and 1nsane had modes where you could just drive around off road as you please, but there's content that's locked until you complete races.
The Saints Row games are on here and while they do have missions you're still free to drive or fly around as you please, but there is traffic to avoid.
Another game is Garry's Mod (only on Steam). You can download maps, cars and plans from the workshop and do whatever you please.

The 'Test Drive Unlimited' series was one where you could just get into a sports car and drive around, without fear of police or needing to get towed. I'd primarily recommend the first game, though, as the second was some kind of online thing, filled with bastards (as they always are).

I've been told that 'Euro Truck Simulator 2' is quite relaxing, if you feel like being a trucker.

I think one can cruise around and have a decent time doing so in 'Driver: San Francisco', though it's been so long since I've played it that I've mostly forgot what it was like. You might get the police on your tail.

Finally, you can just drive around in 'Burnout Paradise', although for some reason I've never done so for particularly long in that game. Something about it just makes me want to get back into the game; it could be the constant barrage of possible races you'll drive past, or suddenly being overtaken by a car you haven't got yet.

In The Crew after having gone through the opening scene of racing to avoid the police stopping you, you can just cruise around, ignoring the calls to race and enjoy the view of American land- and cityscapes.

Jalopy is a game about driving around in an old Trabant-like car through Eastern Europe behind the Iron Curtain. The gameplay element is that the car is quit dilapidated and you have to scrounge engine parts to keep your car functioning, so it's isn't just driving, but there's no need to race. I don't know how much stress keeping the engine running gives, as I've only heard about the game, I haven't played it myself.

(*) The Crew can be found on the Ubisoft Store.

(**) Jalopy can be found on Steam.

(***) I don't know any GOG game suggestions.

(****) I don't know any suggestions for just flying around.

I haven't played Interstate 76/82 in a long time and forgot how they play...

speaking of, anyone wish Carmageddon was in an opwn world and worked more like Burnout/Midnight Club?