Local does a pretty good job of scrubbing private info from the logs and the errors it produces, however there’s always the possibility that something private can come through. Because these are public forums, always review the screenshots you are sharing to make sure there isn’t private info like passwords being displayed.
You can’t grant access but you could edit the Hosts File manually, however Local will need to make changes to this often so it would quickly become a chore and would be better to address the source of the issue so it can work normally.
Are you still getting the “Can’t Update Hosts” errors or something different?
Now that you have MAMP and other stuff removed, maybe it will be worth trying a big refresh.
Export any sites that you have on Local and save them to a folder somewhere on your computer.