what possible modifications could be made to improve the accuracy from a mechanical standpoint to the level of a Saiga or VEPR?
Well, that's a good question. My personal belief is that, as long as the quality of the build is adequate, there's very little difference in mechanical accuracy potential betwen any of the AKMs. If the barrel is in good shape, the crown is clean and straight, there's not a whole lot else to check.
A good trigger helps (you) with accuracy, but an AMD build is going to come with a US-made trigger, probably a Tapco set. And they're usually pretty good. Sometimes, surprisingly good for what it is. (I dropped in a Tapco fcg when I built my AKS-74 and the thing is almost TOO light.)
Finding a solid and repeatable way to benchrest an AMD-65 will help, too. It's a compact gun, with very little in the way of a buttstock to rest on a rear bag, and both a magazine and that oddly angled foregrip in the way of resting it securely on a front rest or bag, as well. Taking the time to figure out or even make up a secure firing platform that can hold that oddly shaped gun will help you make more consistant shots from the bench.
What are your goals? If you simply want to prove to yourself that the gun is capable of a certain level of accuracy before you invest your time into training/practicing with it, this will probably be fine. But there's not much practicality or value in trying to punch tiny groups from a bench with such a gun. It's built to do OTHER things well. If you don't want, need, or plan to take advantage of its characteristic design goals, then owning it might be an exercise in frustration.
If you want a gun that you can stow somewhere inobtrusive, that has reasonable power at fairly close range, and you want to practice "practical" (or "action") type shooting scenarios in various field positions, you'll probably learn to love it.
If you want a gun for plinking from the bench, there are far better choices -- even among the Kalashnikov family (none of which are really at home in that role). In the end, it's kind of like road racing a dragster. Not really meant for this kind of thing.
-Sam