This looks like a fun little steam problem? Either that or there's some metadata issue with the launcher and steam ident?
Make sure forced compat tool is off. Remove the wine prefix from compatdata if exists. Close steam. Reopen. Check if it is trying to download the runtime and stellaris again.
If so, it's steam deciding that the game is better run under WINE. Likely getting a communication error because it's just launched inside the wine prefix.
The way I 'fixed' it is to use 'Steam-Play-None' which you can get via proton-up. There's a github repo for it too. It's just a stub that tells steam to bugger off.
I don't know how steam determines all this, but there might be something Paradox can do here?
Just for reference:
Distro: Arch Linux
Kernel: 6.6.4-zen
Paths: ~/.local/share/Paradox Interactive, ~/.paradoxinteractive
Erroneous wine paths: ~/.wine/drive_c/Program Files/Paradox Interactive, ~/.wine/drive_c/users/user/App Data/Paradox Interactive
I also did not enable steam play initially, it simply decided it's a windows game now.