Wow! I feel some chagrin; it never even occurred to me to try another USB port. There's also an HDMI port so I can dispense with the thunderbolt adapter for my second screen. Yay! Mac for Dummies might be needed here. smh
No anti-virus gunking up the machine, to my knowledge. My backup drive is relatively new, though I purchased it second-hand. It had the original software and I formatted it to use with Time Machine. For that reason, I'm pretty confident it isn't failing. However DriveDX tells me the 'diagnostic of the external drive is unavailable.' ¯\_(ツ)_/¯ Scans of my computer SSD and another external SSD worked fine and the T5 is connected via USB Hub connected to the original port.
I already tried physically removing and reinstalling the TM HDD drive, a 2T Seagate drive showing up as Ultra Slim MT.
Oddly enough, when I ejected the TM drive and moved it to the other port, I wasn't prompted for the password, the problem that set this whole thing in motion in the first place.
The random disk ejection thing was happening well before I connected the Seagate drive btw. The good news is I have not seen that notification in a couple of days.
The bad news is that while checking the troubleshooting pages for DriveDX, I learned that Mac OS doesn't allow multiple drivers for the drive at the same time, and they name my other external by name (Samsung T5 portable). It says I have to uninstall the driver for that drive. There's also a comment on another support page that indicates 'SMART support disabled'.
This is all well above my pay grade. I'm going to cross my fingers and soldier on. As long as my TM backups are occurring without any issues on the ext HDD and my music files are being saved properly to my other T5 SSD, I will assume all is well. If I start seeing the 'disk not ejected properly' message again, I'll revisit.
Thank you so much for your expert input @HWTech!