Time Machine -> reformat drive and reinstall -> Time Machine?

Hi, not sure how to ask this well. I recently reinstalled my old (mid 2014) MacBook Pro and while doing so used case sensitive APFS for a file systems. Installed all my apps and things, some of which required some additional work (pulling things from backup, performing a series of steps to restore the backup etc. - old Day One app, old Things 2 app e.g.). I didn't realize that Steam requires a case-insensitive file structure to install and I wanted to play some old games 🥲 Now:

I could do the whole reinstall again, that is annoying.

Or maybe I can back up everything to an external drive using Time Machine, erase/reformat my Mac and do a clean install and then restore everything from said Time Machine? Is that how the thing works?

Appreciate your time helping me out, hope to hear from you!