It's not nice to criticize when someone asks for help -- but you should have
learned in school that a dead battery means that a device will just sort of
lie there and do nothing.

Why the unit should work properly when being powered by the 5V from the USB
cable, but otherwise produce a buzz, is not clear. How does it work when
being powered by your car's battery (that is, the plug-in cable)?

It's almost certainly not the battery. As you point out, it's difficult to
replace SMDs. I would suggest sending it back to Garmin.