In terms of repairing that specific unit's hardware -- I don't really know. I'd have to spend some time reading through a repair manual. Even if it's not the exact repair manual for this specific model, manuals in general can sometimes tell you all that's needed. After all, VCRs are mass-produced commodity products (even the pro decks) with many similarities.
The MP4 has substantial horizontal signal jitter. It's so bad as to be a vibration of the image. An internal TBC would fix that. Even certain DVD recorders as pass-through may fix that.
I don't hear any distortion in the MPG audio. What I do hear is a lot of hiss, fuzz and gurgles -- and that can be removed, leaving just the good audio.
The MP4 audio's gain is too high. Reduce it with a mixer before digital conversion input. The distortion only happens at loud volume peaks. (There's really nothing that can be done to fix this in software, but I think you already know that.)
This is all fairly standard stuff, nothing too unusual here.