Are there any "captains-only" airports in the U.S.?
As already mentioned in the comments, I don't think there is such a thing as "captains-only" airports. The airline/operator might impose special requirements on crew, such as a certain amount of experience and/or being a captain. It might also require the crew to be additionally trained for that specific airport (this can for instance be conducted during the Operators Proficiency Check).

To add to that, I could be captian on one flight, and be first-officer on the next. Makes no sense to not allow me to conduct the approach/departure, just because my rank is not captain on that specific flight.

In Europe, I know of some airports in the Alps (and Cannes on the French coast, just to name one) that require a yearly theoretical test to be made by pilots operating on the airport. This is due to some very specific rules at those airports.