Your question & solution both would suggest there's some kind of permissions issue on the drive - essentially it doesn't think it 'belongs' to your account.
If you've recently changed machines, or migrated there's a chance it still considers the old machine to be owner [even if the account names are the same]
Try Get Info on the drive & see what it says.
You might see something like this -
Click the lock & authorise if you need to make changes.
"Fetching" is a user from an old machine who wasn't migrated to this machine.
Also not that this drive has both staff & admin groups. staff would be added by the system as the generic group permission, admin is an ACL. Alternatively, the group could be the same name as your user name, but starting with an underscore. e.g. fred (Me) & _fred
This is, in effect, a group containing only (Me).
Ignore ownership is the 'quick' way to solve issues, but may not be available on a Time Machine drive, so
If you want to strip the permissions down to basics, then you need
fred (Me) - Read & Write
staff - Read & Write
everyone - Read only
When set, then use the 'gear' icon to 'Apply to enclosed items...'