13 years after it was installed, a kitchen fixture for an MR 16 bulb started giving trouble recently. The bulb seemed to be dead and when I removed it, one of the pins had turned noticeably gray. Filing some of that gray... corrosion(?)... off the pin resulted in the bulb lighting up again for maybe ten days.

The fixture is one of two. The other one had a problem four years ago that was taken care of by replacing the receptacle that the pins of the bulb plug into. So I replaced the receptacle coming out of this fixture as well. Put a new bulb in. Within half an hour it seemed to go dead and when I removed it, that gray... corrosion(?) had developed on one of the pins.

Does the transformer need to be replaced?

Yes the transformer is defective. May be easier to buy a new track head ,finding a replacement transformer may be difficult with such an old track head.