OK, presumably the machine does play back existing tapes (e.g. a calibration tape) both on sync and repro heads for all channels?
And you've done the obvious thing of making sure that the record heads aren't completely coated in muck?
Are you recording all tracks? And if you monitor off sync during recording you see a signal (the input) but if you monitor off the repro head you get nothing?
If playback is working, but absolutely no tracks are recording, that implies something centralised. On most machines that would be the bias oscillator, but the MX80 manual doesn't say where that is, nor does it proved schematics for the audio cards. It might be doing something weird like having the audio cards generate their own bias signal from a division of the main CPU clock or something.
This is purely a guess, but there is a pad on the Audio Control card, IIRC the 'Bias select' switch. Have you tried both positions for that and the Ref Flux switch, just in case one of those has got fluff in it? (That did happen on my TSR-8 once, I thought channel 8 had died but the level selector switch had just gone weird )
Good luck!