Something doesn't compute here... If you've got an OS on your TM drive - I believe it was ganbustein who, a few years back, posted the option to make a TM drive bootable. - why do you boot into Recovery rather than the OS when you select the drive from an option boot screen?
...Time Machine periodically says it doesn’t have enough room to complete a backup and I start from scratch. i.e. I erase the drive and start a new backup. Most recently was about 3 or 4 weeks ago.
But if you erase the drive, which sounds like the
entire drive, the OS would be erased and need to be reinstalled along with TM's being restarted, and it doesn't sound as if you do that.