Can not mount, erase, or run First Aid on external Time Machine hard drive.

Within Disk Utility you may need to click on "View" and select "Show All Devices" before the physical drive will appear on the left pane of Disk Utility. Select the physical drive and erase it as GUID partition and MacOS Extended (Journaled).

https://support.apple.com/en-us/HT208496

You should not use a TM drive for more than a TM backup. If the data on the other partition is unique, then it needs to have its own backup as well.

In order to fix the issue you will need to erase the whole physical drive and create a new TM backup.

Your only other option would be to run the paid app Disk Warrior to try and fix the TM volume.

I had my machine returned from Apple after getting it fixed for a separate issue (new display) and had asked for a fresh osx install as well. When I got home and tried to run "restore from time machine", I encountered the issue where the time machine external drive was not mounted. It was showing up as greyed out in disk utility, and clicking "mount" did nothing, and clicking "first aid" returned the exact same results as you described above.

I googled a bunch and had read many similar topics where data repair was required and various other steps, but I couldn't risk formatting or running anything on my time machine drive because it has all my work on it.

This will sound crazy, but in disk utility, after clicking on the greyed out time machine drive, to the right of the "mount" and "first aid" buttons, there is an "i" information button. I clicked that randomly to see if it would tell me anything about my drive, and after loading for some time, it displayed a bunch of info (I didn't pay attention to what it said specifically) and incredibly, my time machine was mounted!

I'm now restoring....

I don't know what happened, or why, or what caused this, but I am incredibly relieved. I'm not sure if these were the symptoms you're experiencing but worth a shot?