

First, make sure you are using EXT4 file system on your drive, and it’s on /etc/fstab. Then you could see if mounting your HDD to your main steam library location on your home dir fixes the issue.
It’s always a good idea to check steam logs to get a better idea what could be going on.
The issue tracker is here: https://github.com/ValveSoftware/steam-for-linux/issues
If you filter issues with keywords “external library”, you have 68 open issues to compare to yours. Eg. one is a shader pre-cache issue, disabling shader pre-cache may help. In another one people report needing to run a console command to mount the external library.
Hope this helps.

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.