MacBook Air Monterey, had this issue of ejecting drives while trying to get a Time Machine back up done. Tried many suggestions here, nothing worked. Used disk utility to change my Seagate 1TB external to APFS and now I don't get the disk eject error. Note it backs up much faster now. I'm no where near knowledgeable on these issues just sharing what worked.

Do our external backup drives have to be APFS to be compatible?