I think it will be a new game in the same way that Splatoon 2 is a new game. Clearly built on the game that came before it, but with enough changes and additions and UI differences to really make it stand out as an entirely new separate title, even though at its core it was built upon a port of the Wii U version.
Smash Run definitely should make it to the Switch version, that's a great mode reminiscent of City Trial in Kirby Air Ride which is a mode in a game that I still love to play to this day. They could really work with that and expand the idea and options of Smash Run.
I also really want a proper adventure mode, and think that's likely as well.
Finally beyond more fighters and stages, one thing I'd love to see is a new mode called something like 'Rotation Smash'. The idea is simple, based on your final battle in Smash Tour (which sure bring that back to the Switch version too even if it was no one's favorite), where each one of your stock was a fighter you had acquired during the game. In Rotation Smash, you choose how much stock you're fighting with, and you get to choose a fighter for each stock. You could choose the same fighter every time if you wanted, or a different one for every stock, or alternate between the same 2 or 3 fighters, whatever you choose. I think this is a great idea and something they already gave us a taste of with how Smash Tour would end.