That's it exactly. I suppose we can bang out all the physics we want, but sustain from an instrument is most probably going to make for a less characterful TONE which is what a seasoned player desires more. Acoustic instruments may be a different kettle of fish, but hey.
As a system, an instrument is passive and all frequencies will be dampened as opposed to reinforced. Adding amps and a method of this amp inducing sound to the instrument introduces a loop to the system. Feedback, reinforcement, call it what you will.
When it comes down to it, an instrument which sustains indefinitely or as best it can will by definition not dampen frequencies. That would mean the tone is flat. Characterless. Unless you have singing fingers, voiced picks or timbre-laden strings. Doubtful.
I think what makes instruments sound INTERESTING is down to the selective dampening of wood (filtering and interference if you will) which is in effect going to reduce sustain.
The whole topic has been done down to a tee by better people than us, and it was probably still a bucket of yap.
So yeah, what Greg said! All due respect to the physics heads (which I dabble in when relevant) but I think it's the wrong avenue to pursue! Wrong question, whether there is an answer or not (in my opinion)! :-D