OK, I'll have to eat my words in public.

It was a defective pot - or actually pots - they are two ganged 5k pots. Why my testing showed good I'm not sure, could have been the pressure I put on the pot with my test leads made them make contact inside.
In any case, I gave up for a couple of hours and then I took the board into my living room and looked at it carefully between TV commercials. I then noticed that the pot had a crack in it. I couldn't see it very well when the front panel was on. I went back to the workbench and took to unsoldering the thing. When I pulled it out, it fell apart. The carbon traces were cracked.
I then did a quick and dirty test using four 2.2 k resistors (two per pot). I simply stuck them in the holes where the pot(s) went (I didn't even use solder), simulating the pots in the middle of their rotation. When I kludged the mess back together the frequency did indeed change. The other thing that didn't work before was the VCF input on the back - it wouldn't change the freq. as it should when applying a 0 - 10V input, and now it did. So I was very confident that the problem was solved. Someone smashed the freq. knob against something and broke the pot.
But I then had to find a part to replace it with, and that proved very difficult, as GW Instek will not sell you parts or give you the schematic or parts list. I must have searched for two hours before locating the right size ganged pot in China and Thailand. Since they were only 0.99 cents each, I ordered one each from two different suppliers hoping to get one quicker than the other.
So, again this Ebay find, "for parts or not working", is proving to be a good deal. Here's the pics in case it helps anyone else to fix one of these.
The pot -

This is where the pot was installed -

Main board -

This is the frequency counter board -

This is the front panel -
