Secondly, why the characters act they way they do:
Why does Apocalypse want to kill everybody? Simple. HE'S THE FLIPPING INCARNATION OF EVIL.
Why does Benevolence want to stop him? CAUSE HE'S THE FLIPPING INCARNATION OF GOOD!
Another thing: SPOILER ALERT!!! THIS GIVES AWAY PART OF THE STORY (although it doesn't have that much impact - read it and you'll see) !!! READ AT YOUR OWN RISK!!!Spoiler
Their only motive as a group is to use their powers, really. Benevolence wants to help people (plus the thing with the spoiler above...) and Apocalypse just wants to fight people (plus, Benev won't let him out of his sight).
And about Logic. He's really just a character I added for a few moments. He really doesn't do much, considering that he's a ghost. He's more of a side character that's with Benev and Apoc. HE pretty much has to exist, since the original guy is supposed to be a bit of a genius, and sometimes good, sometimes bad - you see where this is going? Benev = good side of OG (original guy). Apoc = bad OG. Logic = brains of OG.
The bolded would just make the story inherently boring imo (or at least really difficult to write around without making it boring). Especially after a few interactions or battles between the two, I think it would get old very fast. Usually, the hero has a motive beyond just being good, and the villain has a motive beyond just being bad.
Though starting the story off like that and then pulling the twist you mentioned might actually work out very well. Because it is certainly an interesting twist, and the contrast would definitely be greater if they were both 100% good/evil before that.
I still think it would be a good idea to add some relevant plot structure and some plot line staying consistent through the entire story, unless you want this to be a gag manga with fighting/story aspects rather than the other way around.