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?