Did you format your old HDD after you put it in the enclosure? If not, all the old stuff will still be there and will be owned by "root" which means your non-admin user won't be able to write to anywhere on the drive.
More so if you want to have some fun playing with electronics. You can build a Rasberry Pi to control your extra storage via usb through a external drive adapter. Then you essentially have a custom built NAS. Then have the NAS initiate backups from devices connected on the network every day or so.
In essence provided you can set it up correctly when you or your family come home and all their devices connect to your local network, they can all be backed up locally and automatically.