Well, the new version of WoW (forever) is coming out soon and it has native controller support. It's got more horizontal progression so you can play at a slower pace without being left behind. I'm going back after many years away.. I've been thinking about some way to play it on the TV (3 floors lower than the PC...). I see folks playing the beta on the Steam deck but with just the dock you wouldn't get more than 1080p medium settings, so this is very interesting information. My house has separate WiFi access points linked by ethernet over coaxial.

.... So if you aren't using steam link or moonlight, what exactly are you using?

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.

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.

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.

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.

If the PC is close enough, wouldn't it be better to just plug the TV in as a second monitor?

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.

Not answering your question, but related: I mostly use my Steam Deck with a dock, TV, and controller. The only issue I’ve really had is that it doesn’t always run the fan as needed, and then the CPU will throttle when it gets too hot - with more demanding games. It happened a lot with Diablo IV and more recently with Need for Speed Heat or Palworld, less so with games like Stardew Valley, Casette Beasts or Pizza Possum. I bought an external fan that forces it to cool (https://jsaux.com/products/fan-cooler-for-steam-deck-gp0200) and haven’t had any throttling issues since. The noise is a little annoying sometimes though.

Answering your question: the above listed games are fun, and easily played with a controller.

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.

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.

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.

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

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.