When you boot from the Recovery Drive using ⌘R the screen looks like this.

I just checked all the computers here and all are running Mojave. An Option boot does not offer Time Machine as a bootable option. Go to System Preferences > Startup and see if Time Machine shows up there as a bootable option. I have no explanation why yours shows up, but Apple does not believe you can boot from a Time Machine backup either. Are you sure that at some point you didn't install MacOS on your Time Machine drive?

Last edited by joemikeb; 03/03/19 11:34 PM.

If we knew what it was we were doing, it wouldn't be called research, would it?

— Albert Einstein