I guess what it all comes down to -- and this is solely personal opinion -- is whether you think TFF began with a good story that then got lost through bad filmmaking, or whether it was a bad story to begin with that was just made worse by the bad filmmaking.
I personally think the former. I think the story, as constructed, is solid. I think there is a lot of good stuff there. I think it was brought down by three things: the insistence by the studio on including lots of humor in what was supposed to be a serious story; the choice of a visual effects company unqualified for a project of this scale; and the refusal by Paramount to adjust the budget to allow for rewriting and reshoots when the problems were discovered.
Most people focus on re-doing the visual effects. That's certainly possible, and even perhaps not terribly complicated, with today's technology. And I think it would help. But that's not the biggest thing that could be done. The biggest thing would be to re-cut the movie. You could remove a lot of the ridiculous humor moments without removing any key information from the story. Okay, the fall of the mountain and Spock's Superman rescue are pretty tightly tied into the more touching moments of Kirk discussing the fact that he knows he'll die alone. But other elements, such as the oft-cited Scotty banging his head on a bulkhead for example, are not and could be trimmed with ease.
A good, thorough re-cutting of the film could work wonders, IMHO. But that's because I think it's got a good story that it starts with. Your mileage may vary.