What you have probably found is the air pump that helps to maintain pressure in a "pressure-type" rather than "demand-type" system. I don't know precisely when the switch was made to the demand system, but for some reason the year 1964
sticks in my mind - - I know that my '64 Overlander has always had the demand system; and one of my friend's who has a '60 Overlander had a pressure system (at least until very recently). The major difference between the two systems is that the pressure system will have a metal water tank (usually galvanized steel) while the demand system will have a "plastic" tank. Another clue is that the city water connection and water fill apparatus are very often the same device on a coach with a "pressure-type" system; while you will find a separate connection for city water and a filler tube for the fresh tank on most coaches with a "demand-type" system. Also, the pump runs nearly every time a faucet is opened with a demand system, the air pump "typically" will run less frequently.
From your desctiption, it sounds like the pump may not be functioning properly, as I would think that it would be trying to add air to the empty water tank to bring it up to its pressure set-point. If it is like my Overlander, there is a light switch near the kitchen sink (may be inside the kitchen sink cabinet if it is like my friend's '60 Overlander) that controls power to the pump (it looks like a normal household switch).
Good luck with your investigation!