How do I get Time Machine to see a backup of another Mac on the same disk?

I want to force Time Machine to point to the backup of my choosing and control what information is recovered from the backup. How do I get it to see the relevant backup.

I DO NOT want to use Migration Assistant. I need more control of what is copied for security reasons.

Backup of old Mac exists. New Mac purchased and has a known clean OS. I do not want to re-introduce security problems using Migration Assistant.

Currently the backup of the old Mac does not show up in Time Machine. Setting up the disk as a backup disk only creates a new backup file, which is of an empty Mac.

The account names and passwords are intentionally different between computers. But, I still need to migrate files from an old Mac/account to the new one.

There is literally no information on how to enable this. I'm OK with digging into Unix commands to make this happen, but I don't understand how things are architected to be able to do anything meaningfully useful.