That's odd, usually time machine drives are themselves bootable, to allow you to reinstall the os prior to restoring the backup on them.
Am I missing something? AFIK, TM backups are not bootable. You can restore from them via the Recovery Partition or by entering TM while booted from your main drive (partition) but otherwise, that's it.
Jon
macOS 11.7.10, iMac Retina 5K 27-inch, late 2014, 3.5 GHz Intel Core i5, 1 TB fusion drive, 16 GB RAM, Epson SureColor P600, Photoshop CC, Lightroom CC, MS Office 365
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