Time Machine Won't Back Up Root Data

Hi all!

My Macintosh HD - Data disk has been producing a lot of invalid codes so I set out on a mission to prepare a backup through Time Machine in order to erase and potentially fix the Macintosh HD - Data disk. So far on my journey I've managed to successfully use a WD My Passport to perform this backup, but when I go into recovery mode during startup and attempt to reboot with the backup on the WD My Passport I get hit with a new issue:

I took that image from this post here because it is the exact same issue I am having:

https://discussions.apple.com/thread/4618087?sortBy=best

The only difference between this post and that post is about 12 years. The solution to that post no longer seems to work for this issue unless I'm doing something wrong. I'm able to successfully select the WD My Passport as a Time Machine backup, but it would appear that it is not including the Mac OS X system.

If anyone has any knowledge at all of this situation I will be forever grateful if you could help me out.

Thank you!