You and I messaged back and forth about water pumps a couple of years ago. I am the lady who has gone through 6 water pumps in a little over five years. If you remember, what mine does is run fine as long as I use it every day. If I use city water for a few days and then try to use the water pump and work from my tanks, like when I am driving, it refuses to pump water. It takes several minutes of dry-running to get it to produce water, and sometimes it has taken an hour. Mine does not do the cycling yours does, but it will run briefly every couple of hours, so I turn it off at night.
I think both of us have a small air leak in the water system somewhere, but I have had mine checked at the factory, and no one can find anything. In any case, I still have pumps fail about once a year or slightly more often. I have come up with two ways of living with this problem: 1) I will "exercise" my pump every couple of days, shutting off city water and running pump for an hour or so, and 2) I bought a new water pump from CW in Florida last March and bought their extended three-year warranty. What this means is that I pay only labor on getting a new water pump installed, which is what I did last month.
Lousy solution, I know, but I will be interested in hearing if you come up with a permanent solution.
Also interesting that we both have Fleetwood motorhomes.