This is my exact setup at the moment. I've been playing a bunch of Rogue Driving, Elite Dangerous and Succubus Successor: Delilah's Juicy Journey.

I've been playing Final Fantasy XVI like this and it's been perfect. It just became Deck verified very recently but that's a stretch.

The answer is really just any game that is well suited to controller + TV. The aspect of streaming to the Deck is largely irrelevant since, as you mentioned, the experience feels so close to native.

I used this type of setup to pkay Harold Halibut, the Quantic Dream games (Detroit, Beyond Two Souls, Heavy Rain), Elden Ring (+ other soulsborne), and anything else that doesn’t run at its full potential directly on the Deck.

Here's a suggestion that doesn't even begin to address your question, have you considered a long HDMI cable to connect your TV in your lounge to your PC directly? Best of both worlds, comfortable seat in front of the TV, no input delay, play anything your PC can handle without middlemaning through the deck.

Steam has a list of games that are "Controller Friendly". I mean the Library view in Steam client, next to "All Games" list. On your Steam Deck it lists games that are verified for Steam Deck, but on PC client it lists games that are Controller friendly in general.

You can create a custom Collection for that too, a list that is available on all systems you log in. Go to "All Games" view, click "Fliter" (or X button), scroll down to "Hardware support" and enable filter "Full Controller Support". Scroll down and click "Save as Dynamic Collection". Name it and save it. Now this is a dymanic collection, which means it will automatically expand with new added games that fit this criteria. I would say all games from this list should fit your requirement and use case of being playable with a gamepad.

This is the first time I've thought about this, but you've made me think I should try this games like Kao or Bubsy4D or even just R&C Rift Apart.

I streamed Expedition 33 to my deck when it was my shift during the night to look after our newborn whilst they were sleeping.

Honestly can't remember much of it since that period of time was a blur, but what I do know is the streaming worked great, although I had to change a couple of settings as games would sometimes freeze. I think it was something to do with the encoding and decoding, so you may have to look at that if you have the same issue.

The new star Wars racer would probably be dope, or one of the Redout games. You could see if Elden Ring will run with the full graphics instead of the fog version it usually plays at on the deck.

edit: These run natively, I misread your post. Apologies.

Vampire Survivors

Slay the Spire

Children of Morta

Dead Cells

Currently using sunshine/moonlight to my Rog Ally for Dawnwalker. I also use my Xbox with moonlight to stream occasionally. Just about any Unreal engine game runs rough on handhelds due to stuttering, so I use my desktop and stream it. The vast majority of the games I play work great with a controller.

When I was injured civ 6 on a controller was a great time to just sit and play and enjoy.

Megabonk

In my time I have found that as long as the game isn't terrible controls, and isn't extremely latency dependent (you could try enabling Pyrowave codec in steams streaming or try Sunshine to bypass steams streaming) then the real limitation is being able to see the menus from the couch! Lots of modern games get this right, especially if they already have console ports, but sometimes you just can't read the stinking menus!