I think that if Nintendo had been able to make the ideas behind Star Fox 2 work, we would have seen more of them in Star Fox 64. It was probably overly ambitious. If they ever do bring something like that into the Star Fox series, I hope it doesn't preclude the classic gameplay.
However, the idea of things happening while you're not there is certainly a fun one. They could work it into the branching paths of Star Fox 64. For example, if you take one route, you encounter a fleet of supply ships to destroy. If you go a different route, then those ships reach their destination and bolster the forces there, so you'll have a rougher time if you go there. Combine this idea with the branches converging at points along the way, and you have a natural difficulty progression, too. The easy route takes you through the supply convoy, and the hard route takes you through something else, but both lead you to the planet the convoy was heading to. They could also make it so that along a higher difficulty route, you take more steps before getting to a particular planet than you do on the easy route, giving the enemy time to prepare for your arrival, especially if you tip them off that you're coming by failing some objective on the way, like letting some enemies escape, or not destroying all the transmitters at a base (kind of like the search lights on Zoness).
I also like Spak-Spang's idea of multiple runs through the same area. Sure, you could do it in all-range mode, but if the game forces you into a set path, the level can contain better scripted events. Not to mention other changes, like changing the orientation on each pass. You'd probably be in space, after all.
I think the game should use the analog stick. With the analog stick, different vehicles could use the remote differently, in ways that suit them better. Now, I wouldn't be too sad if the next Star Fox only had arwings in it, but the tank was pretty cool. Chasing the train is one of my favorite parts, too. With the analog stick controlling movement, the remote could be used to aim the tank's turret anywhere, which would definitely be an improvement. That would work in Ian's boat level, too (I have absolutely no problem with the submarine being sacrificed to make a boat, incidentally). In flight, moving the remote could choose targets for charged shots and nova bombs, and maybe even the Great Fox's big guns.
You could boost and brake by quickly moving the remote forward or backward, and twisting would replace the L and R buttons (or Z and R), although I don't know about barrel rolls. Twisting twice in a row quickly enough to save yourself by rolling doesn't sound very comfortable. A button modifier wastes a button. Perhaps the game could be changed so that twisting always causes a barrel roll first, then leaves you leaning, but I don't like that idea, especially for the tank. Maybe you could roll by quickly tapping the analog stick in the same direction you were leaning, Smash Bros. style, though that's not ideal, either, since you couldn't roll without moving.
I haven't accounted for all of the available buttons, so they could add a few new options to the mix, too.