My best guess is that you have an GPU that either doesn't support Vulkan, or has driver issues. But we shouldn't guess, that's what logs are for.
For Steam logs, running Steam from terminal as suggested is one way. Do note that error with wrong ELF class for game overlay library when starting any game is normal, since Steam tries to load both 32 and 64 for bit version for each game, and the wrong one will always fail. Arch wiki has more information.
https://wiki.archlinux.org/title/Steam/Troubleshooting#Debugging_Steam
For Proton logs, set environment variable PROTON_LOG=1. You can do it in Steam launch options, see Proton Readme for more info.
https://github.com/ValveSoftware/Proton?tab=readme-ov-file#runtime-config-options
With hardware and firmware issues system logs often point to right direction. Again Arch wiki has a good tutorial on it.
https://wiki.archlinux.org/title/Systemd/Journal#Filtering_output
Games often have their own logging too if you need to go there. You'll need to look those up, as they vary by game.
I hope this helps.
There are basicly two ways to go with regressions: bisecting or research.
With bisecting you restore a working backup, and try to isolate the breaking change. In your case you could try updating one package at time and testing. Since these are often GPU related, start with kernel and mesa. When you find the breaking update, you can either report it on your distros issue tracker, or git bisect it further to the breaking change in the source code to increase the change of it getting fixed quick.
With research, you look into relevant bug reporting databases. These include your distros issue tracker, Valve's issue trackers both for Steam and Proton, DXVK issue tracker, freedesktop.org and kernel issue trackers.
These are a lot of work, so most people just try random stuff. That's why you often get suggestions to do so.
Sorry I don't have an easy fix for you.