I think an HD re-release actually isn't a great idea from a business sense, though I would love the opportunity to play the game on an xbox 360/ps3. Many 3D games from the early 2000's haven't aged very well. I just beat the re-release of Jet Set Radio, and while it was awesome (of course :-P), the controls were pretty brutal and the game is getting lukewarm review scores (gamespot gave it a 4.5!). Most people aren't going to give Shenmue the benefit of the doubt that "flaws can be overlooked because it's from 2001", and for $10-$15 there are tons of great original games on the XBLA/PSN service. I'd definitely be afraid that the HD re-releases wouldn't do so well. As an example, I remember a big criticism of the xbox version of Shenmue II was that it was basically a DC port. How would a DC port be received on the 360/ps3? In a way, it could almost turn people away from SIII. A re-release would be great for the established fanbase, but I don't think it'll really help make too many new fans. Making major changes can be difficult and costly, too.
What I would do is make a series of videos that retells the story using the game engine for SIII and record or rerecord the lines with new voice actors. The videos can be released episodically on gametrailers.com, xbox live marketplace, PSN, steam, etc. and can serve the purpose of getting newcomers hooked on the story and leave them with a cliffhanger. You can tell the story without having to about whether or not newcomers will be turned off by the goofy voice acting, or gameplay/graphics from 2001. That makes the most sense to me and I imagine would be more cost effective than an HD rerelease.
As for SIII changes:
I would re-evaluate the role of money and what you can do with it. It doesn't serve much of a purpose in S1 and S2, except for a few story segments. Nothing you buy really does anything.
I would expand upon the fighting/training system - Require a wider variety of strategies, have more opportunities to spar, have more moves and combos to unlock and learn, etc. Maybe have a level system for individual moves, which unlock additional combos and abilities. Be careful to not make it to "video gamey" though ;-)
In general I think the "critical path" could probably move at a faster pace, with the option to explore at your own pace. Remove the tedious chores like airing out the books, moving crates, or at least have them last for a shorter amount of time. I really loved the Iron Palm/Leaf catching segments in SII on the other hand. I guess because they were more unique and were a part of Ryo's "training" (even if it just concerned the story, like the leaf catching).
I agree with some more side stories/NPC interaction and all that.