screen buttons inoperable, but beep when depressed. Unit is otherwise cold, possibly too cold in the fridge compartment....

torx 20, possibly a T10 too

A multimeter will check for electrical resistance, however inlet valves do have a mechanical component, that sometimes will simply seize, open, closed or in between. In your case, its likely it's seized closed. It will need to be replaced. Its pretty simple, and the parts is about AUD$20.

It does sound reasonable that its the element. A megger test, or an insulation resistance test should confirm this, once confirmed, a replacement part will repair it.