I am trying to decide what type of TV to buy. Can I plug in my Chromecast ultra to run Stadia (like I would for any other TV)?

Yes, as long as the TV has an HDMI input you're good to go.

I know there are side-loading instructions, but that seems really fiddly

Sideloading an app on Android TV or Google TV is as simple as finding a way to download/transfer the apk to the TV, enabling the "Install unknown apps" permission and installing the app itself.

For the Stadia app specifically, once you launch the game then link the controller it works flawlessly, but you won't have party chat and the Stadia menu is currently extremely slow.

And maybe I should buy a Roku TV or some other Smart TV instead.

I have a TCL Roku TV series 5, and I'm very happy with it as a 4K display. The CCU works as expected. I never liked the Roku OS so I got myself a CCwGTV and I was all set.

If it's a CCU, yes, it will work.

I will say that I've sideloaded Stadia on my android TV and found that it just isn't powerful enough to run it smoothly