This might be a weird suggestion but deep rock galactic sort of fits this theme in my mind. I found it to be pretty chill on the lower hazard levels, you can hang out and play mini games in your base ship between missions. Lots of fun multiplayer, upgradeable classes, short missions, funny VA and themes..

Edit - I somehow missed the multiplayer part. For that, try Torchlight 2, it willl run fine on a laptop from 2005.

Single player:

Tactical Breach Wizards

Brilliant writing, awesome characters, and overall extremely well done turn based game. Has a rewind feature that is integrated into gameplay in a way that both makes it more fun and even less stress.

Well done graphics with slightly more than potatoe requirements.

https://store.steampowered.com/app/1043810/Tactical_Breach_Wizards/

Terraria will literally run on your phone and you can set up a world to suit your needs ie. No combat etc.

Valheim is good for a bit of building and exploration.

I also recommend Palworld as others have suggested. Or Valheim, for it's darker grim cousin.

Maybe unconventional recommendation, but FFXIV has an extremely generous, unlimited time free trial. Like hundreds of hours of free trial. Most of it's dungeons and content is designed for roughly 4 players concurrently (tank, healer, 2dps), though every class does lots of damage so nobody is just standing there bored. There's combat of course as an MMO but it's pretty relaxed pace most of the time. But this is a big commitment of course.

Core Keeper is basically Stardew Valley + Minecraft. You kill bosses and stuff but also mostly gather resources and farm and upgrade gear.

Factorio and Satisfactory has multiplayer, if you enjoyed the automation and progression part of Stardew Valley.

Raft you are on a raft at sea — you all work together to upgrade your raft.

I've heard good things about Dinkum and Grounded and Coral Island but haven't tried them yet.

Lovers in Dangerous space time and Pico Park come to mind. Sort of couch co-op vibes but can be online.

Sky: Children of Light (dont do the season pass shit, just play the main story with your friends and then let it be done and it will be fun)

Webfishing (no controller compat)

Flock Around

Of course, imo, Project Ozone 3 for Minecraft Java 1.12.2 is the best chill game for me, everyone on their own island, just tinkering with their shit. I do go into creative mode if we play this way and immediately setup connected portals to everyone's islands so we can all come and chill and watch other's progress if we want (but its such an early progression that we dont do that anymore, you gotta earn the right to connect to the portal network)

Powerwash Simulator.

You Don't Know Jack. There are several packs of trivia and party games, and you can use smartphones as controllers. Available on Steam and reasonably priced.

On edit: I didn't notice the "virtual" part. I'm not sure if you can do that.

Depending on what you enjoy Minecraft on peaceful mode may be an option :)

Raft, but you will die if you walk away too long but there might be settings where this isn’t the case. I have no idea about the controller since I never tried to play that way.

Dome Keeper is a really fun up-to 4-player co-op game about digging for minerals beneath a dome and defending it from creatures on the surface who try to break in every so often.

https://store.steampowered.com/app/1637320/Dome_Keeper/

It's mostly a chill game, but the defense of the dome can be exciting, though only one player needs to actually man the defenses, the rest can continue to mine below ground.

Starbound maybe?

Might not match the hardware and controller requirements since its optimization changes depending on the machine, and the commands are keyboard and mouse-centric so adapting to a 24-keys controller can be hard.

If it's just two of you I enjoyed phogs with a buddy of mine. Technically you pause if you walk away as it takes both people. I'm not sure you'll find many games that fits literally everything on your list but fingers crossed for ya.

If you're a Millennial I highly suggest the "Emily is Away" trilogy.

3 games that are extremely simple, but god damn do they make you feel like a teen/20 something again.

The first game, "Emily is Away" takes place in AOL IM. it's a visual novel but it's very good and allows you to set your avatar, etc just like you did with AOL IM. it's brutal when it gets to the end. the Second game "Emily is Away Too" also takes place in AOL IM but is a bit more difficult as this time there are timing choicse. the third game "Emily is Away <3" takes place on the original Facebook IM. that one is even more difficult. so the difficulty between the games ramps up.

fantastic series. I have all three and I love them. honestly the first game made me cry at the end.

Palworld can be kinda cozy/chill, but there are no real NPCs to hang out with.

I was going to suggest Paleo Pines because it's basically Stardew/Animal Crossing with dinosaurs, but it's singleplayer only.

Tiny Bookshop! Stardew valley vibe but the core gameplay is more relax as you decide the kind of books you store on your shelves and give book recommendations to your client. Very chill

Escape Academy or similar type of escape room/puzzle games might work.

Slay the Spire 2 has a good co-op option. And technically any digital boardgame (ticket to ride for example) would meet your vibes.

Maybe a little more intense, but Rayman Legends/Origins are pretty cozy.

Any of the Lego games would probably work.

Peak has an "easy" setting that just makes it a goofy teamwork based game.

I'm really loving Splatoon Raiders.

It's not out for another 3 days but Big Walk seems to suit all of your requirements and looks like it will be a hoot!

Go Go Town on steam came out recently, co-op available local or online, controller compatible, up to 4 players (aside: you didn't tell us how many players needed, but stardew is 4 so I'm going to assume a 4 max isn't a problem). Not sure on system requirements, might be a little heavier than stardew but it's not all that graphically intense. Had a blast playing it local co-op with the kids.

Gameplay loop is familiar: extract resources to grow your town, unlock higher tiered buildings, eventually automating most of the logistics such that it runs itself like a little town simulator. The game is very casual friendly and forgiving, for example, you can freely just pick up and move a building, no reconstruction or lost costs, you can't really "die", and if you need to step away, shops might run out of inventory and stop producing but they won't death spiral or anything.

Gone home is fun and cozy

I recommend Heave-Ho even though it doesn't meet your second point. Dying is so funny, I often found myself doing it on purpose for a laugh.

Palia is free to play and keeps expanding their game. No problem if you just walk away, no player health meter, just focus to get extra XP from the things you are already doing. Fun for solo, duos or more. Has hunting creatures that are more challenging to track down, basic supply gathering via wood chopping and mining, crafting of all kinds of stuff, farming and ranching too! Crazy they still have it for free.

Witchspire - similar to Palworld, but "witchy" themed

Astroneer i believe fits all your needs