Hmm, I like the idea of an opponent that persists throughout the playthrough. It sounds like it could give Adventure mode way more replay value. The only thing I would recommend to really seal the deal on this rival idea is to incorporate it into gameplay, which I'll elaborate on. In the context of Adventure mode, let's say that the player is given choices of some kind that determine which rival you get. For example, let's say you have to play Adventure mode by choosing which Nintendo world to enter as you progress, and your path through the levels would determine which rival you encountered. For example, let's say you are playing as Mario and you start with Sonic's world. Because of that decision, Sonic becomes your rival, etc etc. Also, for reference, I think this idea would really fly if the levels had a similar layout to Melee's Adventure mode, or maybe even Subspace's levels as opposed to just choosing from the multiplayer's stage list.
But it doesn't end there, let's also say the first level you choose is a character's stage who is unrelated to the character that you are playing as. Like, let's say you pick Link's stage while playing as Mario. I know Nintendo could easily whip up some kind of rivalry between them for the sake of the feature, but let's say Link doesn't become your rival because he has no clear connection to Mario. In that case, maybe you would fight a nemesis at the end instead...Or, to really top things off, let's say the hardest difficulty is an Adventure mode with both a rival(s) and a nemesis. It doesn't have to be a normal character either, I imagine they could easily throw a boss in there.
Then, in order to spice things up even further, lets say that you can choose your allies as well. This could mean choosing characters à la Subspace, or having an ally/allies assigned to you automatically(as first seen in Smash 64 in the Giant DK battle). The allies could come in for one battle or maybe they could stick around. I think a mix of both would be interesting.
As for adding cutscenes in to sell the "story" idea, I could go for something simple. For example, I wouldn't mind cutscenes containing still pictures of the characters with text describing what's going on in the adventure. Kind of like a comic. It could be boring for some people, and in order to keep the pace brisk like it was in Melee, then I suppose it's better to have some quick in-game cutscenes.
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Just an idea.