When I think beat'em up, I think Streets of Rage (co-op), Double Dragon (co-op in it's best versions), Final Fight (co-op), Castle Crashers (co-op). I'd say some of the prime examples of the genre fundamentally allow for two players to play together, and, at their best they utilize the co-op to allow for team moves, etc... Note that the video review compares it to that genre of games and lists some of those above by name.
So yeah, I'd say it's fundamental feature missing (for this genre, not for every game type ever) to leave co-op out. The fact that they try and patch it up with a half done version of co-op where you can play just the mini games shows that they know there'd be appeal for that feature. I don't think it's an unfair complaint or comparison to say that a beat'em up should be playable by more than one player at a time (much like a shooter, though I wasn't making that comparison at all).
It's also kind of silly to rebuff my complaint by saying I'm trying to make the game something that it's not. Fundamentally, any complaint about a game missing a feature one would like to see is a complaint seeking to make the game something that it's not. I just don't think it's a huge extrapolation to expect a game labeled as a beat'em up to bring meaningful multi-player to the table.
And yes, other systems have games with short stories, and I'd make the same complaint of them. But it seems like an easier fix when they throw some online co-op, DLC, or other way to extend play time on the 360 or PS3. Something that it's a little harder to do with the Wii.