Thanks for the suggestions, I'll look at doing the simpler ones in the morning.
Just so we have the symptoms right: the system worked, I filled the tanks, the system stopped working. All in the span of and hour -- it took that long to fill the tanks from the dock hose. Only after it stopped working did I mess about with closing/opening valves to individual tanks.
Oh, and the backup pump is an identical electrical pump. You throw 2 "Y" valves (inlet, outlet) and a switch (power the other one) to operate the alternate pump instead. I didn't mess about with that until after there was a problem. The filter going into this pump/valve arrangement is pretty clean. (The pumps are immediately under the dinette seat, well above the tanks.) Opening all the faucets didn't work.
I'm thinking it's the valves, as you suggest, so I'll look to make sure they are open. (Maybe what I think is open is really closed.) It sure seems like that's the issue, even though no change was made after filling. Then perhaps it's not priming or something. So I'll try the "open inlet port" second, then the "disconnect outlet side of pump" method third.
One last thing to be 100% clear on the symptoms. When filling, I couldn't get one of the 4 filler ports to open. [I even tried putting a piece of dry ice on the screw-cap (to shrink it) but that didn't work. Having learned from mistakes before, that nothing on a boat should be forced too hard lest it break.] So really one of the tanks was not filled.