Long-time Ubuntu user finally making an account here to ask about this:

Family member was dual-booting Ubuntu 18.04 (ssd with MBR) + Win 10 (slow hdd with UEFI) for years on a ~5-year old desktop (Asus M32CD) when suddenly two weeks ago after upgrading the GPU the BIOS stopped displaying, making choosing a different OS at startup impossible without taking the video card out.

After hours of troubleshooting we installed Ubuntu 20.04 onto a second ssd this time with GPT/EFI partitioning rather than MBR to try to force GRUB to see both Ubuntu and Windows; this didn't work so we just focused instead on getting the most commonly-used programs working on Ubuntu and not booting into Windows anymore. I'm planning to make a separate post to ask about restoring dual-boot.

For a week this worked well, but today when trying to back up personal data to the external USB backup drive (MyPassport 1Tb HDD, very old) new "Input/Output error [readdir]" errors started to show up for random folders on the backup drive, and those folders were found to be empty (previously full). Thankfully we were able to delete those folders and manually copy them over again from the main drive without errors, but I'm not sure what this means. Which is more likely:
- Backup drive is reaching end of lifespan, replace with external ssd (gnome-disks found no errors though)
- Something about Ubuntu 20.04 changed how USB devices are mounted and I need to change permissions (current mount options are nosuid,nodev,nofail,noauto,x-gvfs-show)
- Perhaps I broke something while upgrading the GPU (an old 950 gtx I stopped using in another rig) and now either the mobo or the SATA/USB cables are damaged (would expect a lot more frequent errors/inability to mount drives if that was the case)

Other weird things:
- MyPassport keeps changing ownership randomly between the main user and root
- Nautilus hangs and crashes when trying to open the Trash to clear it (total disk usage is only 1/4 though)

Hopefully this wasn't TMI/somewhat useful; what are the next steps I should take to figure this out?

Welcome to forums!

Some USB adapters do not let diagnostics tools to work, you better take this drive and connect directly to a SATA port, then run where sdX is the name your OS assigned to it, probably sdb. It will take time, but once it is finished you can use smartctl -a to see the results.

Thanks for the suggestion; this is an external backup drive so I'd have to remove the case before hooking it up to SATA; would rather check drive mount settings first before doing that.

Were there any changes going from Ubuntu 18.04 to 20.04 for removable drive permissions?

You won't get any I/O errors due to incorrect mount options. What you describe is hardware failure of some sort. Probably the drive itself, but USB adapter and cables are also involved.

Edit: if you really want to see all mount options in force simply run 'mount', it will print all mounts with their active options.

I agree considering the backup drive itself is at least 4 years old; the only thing that makes me question that is this only happened after installing the new GPU, so it is either a coincidence or perhaps I dislodged/bent the cable connecting the USB port to the motherboard.

Is there a way to run the SMART diagnostics or do a benchmark over USB? Alternatively a way to test the USB port and/or SATA connectors on the motherboard directly?

We won't get anywhere if you fail to read what has been already written.

It is your drive, your adapter, and only you can try and see if diagnostics are possible thru this particular adapter.

Does the gpu your inserted consume more power then the original, power supply maybe? Is the gpu seated good have a good connections?