If it were up to me, I'd just start off with a JL movie in a universe where superheroes are already commonplace and the team is already established -- maybe go the route of many X-Men
adaptations where a new team member (Cyborg? Flash? Zatanna?) is the viewpoint character being brought into the team, meeting the other members, and learning the ropes. I don't think you need to worry about the lack of "common origin" if you just set up the universe that way from the start -- the common origin is that they all exist in a universe where superheroes of all stripes and archetypes are everywhere, a mashup world like the ones from Monsters vs. Aliens, Who Framed Roger Rabbit, Wreck-it Ralph
, and the like where all sorts of diverse characters of a certain type just happen to coexist as a fait accompli. Instead of trying to establish a bunch of individual hero characters whose existence in a fairly normal world needs to be rationalized and justified one by one, you just follow the lead of those movies by establishing the world itself
as one where it's natural for such a diverse assortment of fantasy characters to exist. Once the audience accepts that about the world, then it isn't necessary to go to great lengths to justify every individual character's existence, and you can just get on with the story.