I’m almost certain it is not a programming/assignment issue - nearly every program is set up to use the mod wheel, so unless someone went in and changed every sound to NOT respond to the mod wheel, it should have some effect. And it sounds like your programming test should have worked.
It’s definitely not a battery issue - the purpose of the battery is only to retain memory when the SQ-80 is switched off, and it has no function whatsoever while the power is on.
That means it’s a hardware issue, and it’s likely something very minor. There’s a potentiometer, and there are wires that connect it, and that’s it. So first thing to check is that the wiring harness is plugged in correctly and that there is not a broken wire/solder joint at the wheel end. Here’s a pic of the whole assembly:
So make sure that brown connector is seated correctly where it plugs in, and make sure there are good connections where the wires solder onto the mod wheel pot. My guess is that this is where the trouble lies.
If not, it’s possible the pot itself could be bad or severely dirty, but if this were the case, you would likely get sporadic results rather than no results from moving the wheel. And if this is the case, a spray of DeOxit could bring it back to life.