Interesting discussion about JC'ing and coop. Whether you utilize JC'ing or not, I do think that's an important aspect that sets Devil May Cry's combat apart from other character action games. I also don't think JC'ing specifically would present a problem in coop. Rather, if you did a coop DMC game, there would inherently be an element of competitiveness no matter what.
Yuri (the guy I've been streaming with) and I talk about the concept of "competitive coop" a lot, actually. If the objective in these games is to keep your combo string going as long and as stylishly as possible, then the addition of a second player with that same objective inherently presents a threat to you achieving your objective, unless you work in extremely tight cooperation with the other player. Imagine your pal, playing as Dante, using the Demon Pull to yank an enemy away from you just as you're about to land the final hit in an air combo. Maddening!
That's not to say it's a bad idea--it kind of sounds like the seed of a great idea. Mortal Kombat: Shaolin Monks actually pulled off really fun coop stylish action. You could toss enemies from one player to another and collaborate together like that. For the curious, other games like the JPN and EU-only Oz have experimented with similar "pass the dude" mechanics, but with kind of limited results.