Thanks Mike. It'll be a couple of days before I have time to get back into it, but I'll follow your suggestions, starting with the pump.
This machine is incredibily clean inside and out, but I haven't looked inside any of the hoses. The person I bought it from said it was meticulously cleaned after each use. Absolutely no debris came out of the agitator bars when I tested it.
Pump 3 pushes water strongly and it comes out evenly through the agitator bar for the brief time that it operates (10 to 30 seconds) when I first switch on the unit. That suggests to me that the problem will be in the pump itself, and not a blockage or buildup of crud in the tubing.
I've found several aquarium stores that sell rebuild parts for the Iwaki MD-6 pump, including o-rings, impellers, front covers, bearing cups, etc. If the problem is in the pump, I suspect I will be able to rebuild it. If not, I'll contact Iwaki America and see if that pump is still available.
I haven't been able to find an MD-6 (actually MD-6-N) pump for sale anywhere. It may be discontinued. But the Iwaki MD-10 has the exact same dimensions, form factor, and bracket location as the MD-6 (looks like the same body, from the drawings on the Iwaki site), and it has the same hose connection types and sizes. It has a higher flow rate than the MD-6, so I'd have to restrict the flow on the output side to match the MD-6 if I end up having to completely replace the pump.
I have the W/D module and the replenisher for this unit too. I haven't checked out either of them yet.