Yes, Battle.net runs flawlessly in Steam, have zero issues playing D2R and D4 this way. In fact D4 ran flawlessly on Linux on Day 1, which came as a bit of a surprise to me.

The only thing I still can't run is League of Legends. I hope someone can figure that out somehow

Lutris doesn't have to mean Wine. It can launch apps via Proton too (and so can Heroic). I think it uses https://github.com/Open-Wine-Components/umu-launcher behind the scenes, not sure, but so far I haven't run into any issues with this approach.

This is exactly what I do, works great. I may have had to try one or two different recent versions of Proton via the drop-down, but it's way less hassle than when I used Lutris.

I played WoW for several years this way without much issue. The only tweaking I ended up doing manually was to try to figure it out how to get the game to let me enable ray tracing, which I did get working but ended to being too much of a performance hit for my to leave on most of the time anyhow.

This is what I do, it feels weird, but works perfectly and is the least effort solution