Add write permissions to a Time Machine drive?
Solution 1:
I don't know what exactly you want to do, and most importantly I don't recommend doing it, but it's entirely possible to create directories in an APFS Time Machine volume on Monterey and write to them. Just do it from the terminal:
Create a Folder
sudo mkdir /Volumes/Name_of_the_TM_Save/My_Folder
Then unhide this folder
sudo chflags -R nohidden /Volumes/Name_of_the_TM_Save/My_Folder
Then you appropriate it:
sudo chown -R your_name:staff /Volumes/Name_of_the_TM_Save/My_Folder
And that's all.
But what can be the interest ????? And the risk of "breaking" Time Machine.
The best solution is as @Tetsujin indicates to create if necessary a new volume in the APFS container supporting the TM backup.