I would love "normal linux user remote desktop session suitable for gaming using the host's GPU".

Nothing seems to do exactly that, though this (moonshine) seems like it might be the closest yet.

Can it run a desktop environment as the "Application" that it's streaming?

I've been running Sunshine on my Linux desktop for a while now and connecting to it with a SteamOS handheld and my laptop for mostly gaming for a while now. From what I can tell, the biggest difference with Sunshine and Moonshine when running on Linux is that Sunshine doesn't use a separate desktop instance per stream; by default, the streams will be mirrored to the clients, although there's a way to work around that somewhat by setting up a fake display output device with the boot parameters and a custom EDID with the specs you want (there's at least once repo where someone has a large set of them dumped from thousands of different monitor brands/models) and then setting up Sunshine to output when mirroring, which effectively means that only the remote client will actually have anything displayed.

It took a fair bit of iteration to get my setup to the point where it mostly works the way I like, but it's been pretty stable for months now. I game exclusively on my handheld streamed from my desktop, and on my local network the input lag is pretty consistently low enough that I don't notice any difference from when I had tried to play things locally on SteamOS, only I can easily max the graphics settings on most games I play (and on the rare ones that do struggle, there's often not a ton of downside to lowering the resolution a bit due to the handheld only being 1200p).

I sometimes ssh into my box with a port forward and starting wayvnc, then connect via tigervnc, I only played a few older/simpler games like this on Steam for a short test, but seemed to work. The server was Arch with Sway and client Fedora with Sway.