When you boot from the Recovery Drive using ⌘R the screen looks like
this.
Yup….that’s exactly what I saw.
Go to System Preferences > Startup and see if Time Machine shows up there as a bootable option.
Nope….it only ever shows my main drive or my Super Duper drive, if I have it turned on. However, if all drives are turned on and I use the Option key on a restart, I get Main Drive, Super Duper drive and Time Machine drive.
Are you sure that at some point you didn't install MacOS on your Time Machine drive?
I have no idea….but I guess I must, and I assume that happens every time with my Time Machine. That’s because 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.
Wait, wait, wait....I can confirm that I do have the system on the Time Machine drive. Your query caused me to check which things were excluded under "Options" and the System folder is not among them.