How to make time machine backups that will be accessible from other machines?

Question

I'm trying to find a way to create Time Machine backups that I can still browse in the future from machines that are not the one that created the backup (or from other users). My main issue it seems is with permissions, but I couldn't find any good guidance online valid for the way TM works since the switch to APFS.

I recently got a new mac and noticed that backups created on my old MacBook are not accessible (red icon indicating lack of permissions). My old MacBook was running Ventura (still an intel machine from 2017). The external drive is APFS formatted. How can I make sure when creating backups on an APFS TM volume that I can browse them in the future.

A few details for context:
I did not use Migration Assistant to transfer the information over from the Time Machine volume to my new MacBook. Every once in a while I like to start fresh and use the move to a new machine as a welcome opportunity to do a spring-cleaning, reviewing all my settings and data, manually recreate settings, discard stuff that I don't use anymore etc. That also means moving my files first to in a separate "Transferred" folder and then step by step integrating them back into Documents, Pictures, Music, etc. So that's what I did this time. I set up my new MacBook without any migrating. But when it came to accessing the files from the backup I noticed that since I'm on a new machine, new user, I don't have permissions for those files. And because it's the new APFS TM volume, which is read-only for normal humans, I can't change permissions either. The workaround I did was to use Migration Assistant to create a new user on my new machine, then move all files I wanted from that user to the "Shared" folder, from there move them to the new main user that I had already set up, and then delete the migrated user (there's unfortunately no way to use Migration Assistant for only moving files, it always forces you to copy the library folder over).

Now I'm hoping to either create backups in such a way that I can access them from any machine/user or find an easy way of accessing them a posteriori. The use case is not necessarily migrating to a new machine (there are workarounds like the one described above), but rather the opportunity to occasionally browse past disks, whether to retrieve an old file, or just for nostalgic purposes, how did I organize my files 10 years ago?, what programs did I use?, what was I working on?, etc. I still have old backups from 2008 (all contained in sparsebundles) that I occasionally access. I always encrypt my external hard drives with a password, so in terms of data safety, I don't need the additional layer of protection with permissions (which can be circumvented by admin accounts anyways).

Any suggestions of how to make TM backups nowadays that will remain accessible from any machine/user? I don't need to be able to "browse" them through time machine's interface (which hasn't worked reliably for very old backups anyways). In fact I would prefer to have the option access them as I used to through the Finder's folder structure – as long as I have permissions to do so.

• nohillside: Do the answers to apple.stackexchange.com/questions/119705/… help?
• jan: @nohillside – well, the thing is, at the moment I have not set up Time Machine on my new machine, so the Time Machine icon (with the option for browsing other disks) isn't there yet – I can't seem to figure out how to get it without setting up Time Machine first (for which I wanted to first figure out the best option, hence my question here). But either way, I would prefer not to have to use the Time Machine interface to access data on previous disks.
• jan: @nohillside I'm back to trying to figuring this out. I went through the options in your linked answer. I have set up TM and am able to select "Browse other backup disks ..." – while this does bring up the TM interface with the backups from the previous Mac and even shows me all the existing backups, while I am in the home folder, all subfolder (Desktop, Documents, etc.) remain inaccessible with the red icon – so exactly like when browsing from Finder. And there is no way to change any permissions via Get Info (even the lock is greyed out). Any other ideas?
• nohillside: Not really. I probably would open Terminal and try my luck there.
• jan: @nohillside I did try, but none of the previously described methods using chmod etc. work. These new APFS volumes seem to be completely sealed off from any user intervention.