Time Machine Quit Recognizing my Seagate One Touch

Started using a Seagate One Touch 5TB drive in February of 2023 as a Time Machine backup drive. It worked fine for a few months, then wouldn't work - it wouldn't appear on the desk top and Time Machine said the disk was unavailable.

Using Disk Utility I reformatted it and was able to use Time Machine to back up my computer to it. Again it worked for a few months followed by the same issue.

Seagate support says that the issue is with my computer b/c Disk Utility shows that the external drive is mounted and functional. Yet Time Machine doesn't recognize that it's available. They want me to verify if my computer is the problem. They think the drive is fine.

I use a different Seagate external drive for storing video files. It connects, appears on the desktop, and I'm able to save to it. I don't use it with Time Machine.

Using a MacBook Air M1 purchased in late 2021.

Ventura 13.4.1

How do I figure out if it's an error on my computer?