First Aid fails on multiple items in Disk Utility (Recovery Mode)

You must run first aid on the volumes first, then the container.

If you didn’t do it in that order, try it first before erase and install.

You’ll install on Macintosh HD.

Here are the steps to follow. You likely won’t be able to accomplish the main focus of the article, but there are links at the bottom for how to erase the drive.

Erase all content and settings on Mac - Apple Support

Starting in Catalina macOS 10.15 and above, TM Backup will only be able to Restore your Personal Data and does not contain the Full Operating System.

If you what to be certain to stay only on Monterey without the possible Upgrade, upGrade and eventually reaching Monterey - this is a method but please do not enter this next suggestion lightly.

Scorched Earth Method 

This method will WIPE ALL Data and there is No Recovery - Period.

>> Only works on Intel Based Apple Computer

To perform this action will require booting from a Bootable Installer. 

The Bootable Installer can Only be performed on an Apple Computer 

This will have to be performed from a Qualifying Computer to run the version of macOS to be made on the Bootable Installer. Example : Bootable Installer of Big Sur would have to be done on a computer that Qualifies to run Big Sur.

Notation: If the computer being used to perform this action is Too New or Too Old to qualify to run the version of macOS - this computer can not be used.

Alternative is, to gain access to a Qualifying Apple Computer from a family member, friend or associate.

Extra Special Notation regarding the Touch ID equipped Apple Computer.

About Startup Security Utility and Must Enable from Recovery Mode the ability to boot from External Drive Before Attempting 

1 - Shutdown computer and disconnect all external drive Except the newly created Bootable Installer.

2- Restart and immediately hold the OPTION key until the Startup Manager appears and choose the USB Drive. 

3 - It will present options >> Disk Utilities >> View >> View ALL attached Drives. 

4 - Choose the Upper Most Drive ( not the volumes indented and list below ).

5 - The drive normally is called Apple Media or Apple SSD - that is the drive to Erase and format as APFS with the GUID Partition Map. This applies to macOS 10.14 Mojave and above. 

6 - Once that is done >> backup out of Disk Utilities and choose install macOS. 

7 - Follow the prompts and it may automatically reboot several time. 

8 - Upon a final reboot - Setup Assist will present with the newer version of macOS.

EDITED - spelling errors