Long-time lurker (seen this site go through multiple URL changes over the past 14 years or so), first-time poster.
One thing I've often thought about when fantasising about a Shenmue III is how it would play. I know we all dream of it running on the most powerful hardware imaginable to fully realise Yu Suzuki's vision (apart from the times we dream of it being released for the Dreamcast ), but I've always thought that Shenmue games should push gameplay mechanics themselves as well.
I love the QTEs in Shenmue I and II, but even back then I thought it'd be great if there was a more dynamic input scheme that allowed you to perform martial arts moves and the like on the fly. I thought maybe the Wii was the answer to that problem, but I was woefully mistaken. The Kinect... does anyone even use it for anything other than switching their Xbox on and Netflix? I'm not saying motion controls are the answer, but wouldn't it be awesome if Shenmue III had you performing moves in semi-cutscenes that were not preordained by the game designers? I guess its usefulness might be limited to martial arts masters... Still, something other than a flashing button on a screen seems appropriate.
Dialogue trees are another one; having a Mass Effect-style multiple-choice dialogue tree for each conversation would make it feel more interactive and dynamic as opposed to just randomly going up to NPCs and pressing A to advance the plot.
Shenmue II introduced fast travel and time-skipping, which I think were great additions (the few additions I'd like to see to a remastered Shenmue I).
Maybe a more advanced pet simulator than the cat in Shenmue I? More nuanced sparring in dojos would be cool, one idea I just had would be to visit various dojos/martial arts schools and train with the students there to specifically train certain moves/skill-sets.
Return of the bicycle?
More variety in odd jobs to get money? More types of gambling?
What would you like to see?