If I had to guess, I would say Shenmue III would have been similar to the first two games in the series in terms of budget and structure. Some still might have had an issue with the overall story, but nobody would be complaining about the game's production values. However, if there is one thing that I appreciate about the remasters, it's realizing that Shenmue was simply too powerful for the Dreamcast to handle.
While the performance on the Dreamcast was sufficient, it didn't change the fact that it suffers from some frame rate drops now and then. Just play Shenmue II on the Dreamcast versus the Shenmue II remaster and you will see what I mean.
In the end, I think Shenmue II was immediately followed up by Shenmue III all the extras from Shenmue I & II would have been carried over to III. The game would have the same budget and structure as the first two games and perhaps some of its technical issues as well.