Only Dark Souls 3 refuses to recognize any controller.
I'm also having this issue on the same Linux Mint Cinnamon 22.2 distribution, with the latest Steam client build.
My PowerA Xbox Series X controller refuses to be recognised by Elden Ring or Dark Souls III, however it is recognised perfectly fine as a Generic gamepad by Nier: Automata, Dark Souls II and Elden Ring: Nightreign.
I've tried numerous troubleshooting issues, additionally including uninstalling and reinstalling various Kernel drivers and reinstalling Elden Ring and DS3.
I have also noted the same as the original reporter, in that when the controller is plugged in I can click to edit the layout but on Elden Ring and DS3 the window is completely blank as shown in Fig 1. Nier: Automata controller configuration window in Fig 2 for comparison.
Nothing extracted yet.
I recently switched to Linux mint Cinnamon 22.2 and my steam controller works fine when playing Monster Hunter World and Elden Ring Nightreign but for some reason Dark Souls 3 says that my Steam controller isn't supported when I try to use it for that game. It's extra weird I have that issue since ER Nightreign is by the same devs as Dark Souls 3. I also tried using a PS4 controller to see if that works but again Dark Souls 3 says that the controller is unsupported so it seems to be a Dark Souls 3 specific issue. I tried everything that was suggested in the trouble shooting suggestions but none of it seems to have worked. I tried to reconfigure the controller but that didn't work either. I have steam downloaded through the software manager that's offered on Linux Mint instead of manually downloading and installing from the steam website if that's necessary to know. As for how to reproduce it I wouldn't know. If it's important to know, I have steam client installed onto my computer SSD but my steam games download and install onto a separate HDD that's meant to be used for storage.
EDIT: I noticed that while my controller is plugged in (I use a wire not dongle) and I click on edit layout for the controller on the library page, on ER Nightreign and Monster Hunter World it shows me different layouts to choose on the screen that pops up but with Dark souls 3 it just gives me a blank screen that doesn't load anything.