Some cars also have a garage door opener module in the rear view mirror as well, but I think that is a separate module in its own plastic package. It will have buttons that are accessible on the bottom of the mirror if this is the case.

I guess it's sort of the luck of the draw as to whether the spring shorts on the PCB or not.

I fixed a couple of these with the broken plastic retainers on the bezel and they did not short out.

It's really unnerving driving a car with those springs broken.

It's likely that you've got more than the burnt resistor wrong with the board. I seem to recall that the PCB is not detachable from the mirror back. If that's the case, I don't know how you could repair it even if you had a schematic.

You could just put it back together with the connector disconnected.

No more auto-dimming.