Flutter desktop works shockingly well on Linux desktop in my experience. It's AOT compiled to native byte code, so it's pretty fast. I think the main weakness would be 3D applications, but I can't say for certain.
Cool. I've been waiting for this since Flutter has been first announced. I will surely give it a try. I just hope you don't need to be fluent in Photoshop, CSS and a pack of libraries to create a basic table-and-buttons desktop app in it like that is with the web stack.
No, it's actually pretty fun writing UI's in Flutter, because it's relatively straight forward.
Edit: Fair warning, desktop development code is _interpreted_ by the VM when doing debug builds, so that you can have hot reloading, which is great. But it may seem like your app is slow until you do a release build. Though it seems obvious in retrospect, it took me by surprise at first.