Please see the following WINEHQ error report for system information an log: https://bugs.winehq.org/show_bug.cgi?id=46736
Additional link to Steam Community fixes, which does not work: https://steamcommunity.com/app/208200/discussions/0/846958724796395601/?ctp=27
Just out of interest RBDoom3-BFG works perfectly.
Consider compiling the latest master rather than running Doom 3 through Wine/Proton. Your RX580 shouldn't have any issue - my R9 285 can handle it.
There's also dhewm 3 for the non-bfg/vanilla edition of doom 3:
(In reply to andrew.m.mcmahon from comment #2)
> Just out of interest RBDoom3-BFG works perfectly.
> Consider compiling the latest master rather than running Doom 3 through
> Wine/Proton. Your RX580 shouldn't have any issue - my R9 285 can handle it.
> There's also dhewm 3 for the non-bfg/vanilla edition of doom 3:
Thanks for already known option.
Still - it does not change the fact that something is wrong with mesa driver, that causes wine breaking and therefore game to be unplayable. That means that until this is fixed, this is an Regression.
Also, if you are saying - between the lines - that Mesa devs do not really care as long as there is Linux alternative, then it must be put like this in clear and straight English so that Community is not hoping empty dreams.
You have reported nothing that suggests this is a driver bug. You seem to be suggesting it is because of a steam forum  comment from 2013 about the windows Catalyst drivers.
If this is a regression you need to provide information such as, the Mesa version it was working on vs the Mesa version it stopped working on.
I've remove the regression tag from the bug title as I don't think this ever worked on Mesa and have sent a patch  that should work around some game bugs in future Mesa releases.
Should be fixed by:
commit 1af72fa4d665b9847dff9b22d7a7dea01c0960c7 (HEAD -> master, origin/master, origin/HEAD)
Author: Timothy Arceri <email@example.com>
Date: Fri May 3 13:59:05 2019 +1000
util/drirc: add workarounds for bugs in Doom 3: BFG
This makes the game playable on radeonsi.
Cc: "19.0" "19.1" <firstname.lastname@example.org>
Reviewed-by: Samuel Pitoiset <email@example.com>