I have been experiencing this same issue in Linux Mint Cinnamon, MATE, and Xfce. When I first install Mint on my laptop everything works perfectly fine. After I restore my home folder backup using grsync, my computer can no longer detect sound devices, the icon does not show up on the taskbar, and all other answers I've found online about resetting PulseAudio and similar do not work.
Logging in as a different user fixes the issue, so obviously some config files are getting botched in this process.
I first solved the issue by manually restoring all non-hidden files in my backup, and cherry-picking the hidden ones (browser configs and the like), being careful not to delete/overwrite anything else. I recently reinstalled my OS and deleted all hidden files from my backup that I didn't explicitly want (i.e. nothing related to audio remained). I then overwrote my home folder partition with the backup assuming all configs would be restored to defaults when I booted up again. They were...except my audio is broken again.
TL;DR When I'm very careful about restoring my home folder backup nothing goes wrong, but if I restore all of my old config files or delete the new ones (letting them regenerate to the defaults) my audio breaks.
Is there anything I can try to restore my audio, aside from reinstalling again?