OS Version: MacOS Ventura 13.2

I got a new MacBook and before I sold my previous MacBook, I created a backup through TM.

Now on the new MacBook I wanted to access the backup of my previous MacBook and faced ‘Permissions denied’ issues. Digging around, I found out that my previous MacBook had only 1 user (UID 501) and my new MacBook already has two users and personal one has UID 502.
I already tried out to use the first user to access the backup and it works – But that’s not a long-term solution.

Found How can change broken file permissions of Time Machine backups? and tried to align that to the APFS formatted TM HDD. Unfortunately without success – While I could do the changes in /Volumes/USB_DRIVE_NAME/2023-02-06-162553.previous/ I’ve seen later that the TM backups are mounted read-only in /Volumes/.timemachine/SOME_UUID/2023-02-06-162553.backup and this is where the TM browser reads from.

Is there any way to fix the permissions for my old TM backup so I can access it again? Maybe somehow possible to mount the 2023-02-06-162553.backup read-write?