Surely, the most fundamental first move is to take a meter to the power
supply, and see if there is at least some supply to the system control
micro etc ? The fact that it's covered in grease may well be neither here
nor there as far as the fault goes. Oils and greases are not known for
being particularly conductive or corrosive. Dead is a positive problem,
and one to which basic faultfinding techniques should be first applied,
before going off on an 'if-then-maybe' exercise ...

I'd normally agree, but the fact the unit /slowly/ declined suggests that
something other than a voltage loss is going on.

Of course, it takes just a few seconds to check the voltages. Couldn't hurt.