You have a good idea for political chaos.
If Aerys dies right there, his Small Council is in a bit of a bind – Rhaegar is technically heir, but they hate him and vice versa. Viserys would be better for them politically, but Rhaegar is too well-established to be gotten rid of. It would mean civil war with Dorne backing him (they’re pissed at Rhaegar for cheating on Elia, but not to the extent of kicking Elia’s son out of the succession), and at least four of the Lords Paramount not feeling particularly kind to either side.
If Lord Merryweather is feeling particularly stupid, he tries to hold Brandon hostage for the military support of Houses Stark and Tully. If he’s smart, he releases Brandon Stark into his father’s custody. Indeed, you might have an interesting scenario where Merryweather et al. defect into the Southron Alliance on the grounds that a child king would mandate exactly the kind of ruling council that the Southron Alliance seems to want.
However, Merryweather might simply give it up at that point, accept Rhaegar as King. At which point, things are pretty shaky – the new King is an accused rapist and kidnapper who’s just alienated House Stark and House Baratheon (who have powerful allies) and the Martells aren’t too happy either, rebellions have been fought over just this sort of thing.
So at a minimum, Rhaegar’s going to have to do some very fancy footwork. Returning Lyanna is tricky – she’s going to be seen as damaged goods. Announcing they’re married is also tricky – House Stark might be slightly mollified but not much, House Martell is going to be enraged if Elia is put aside, House Baratheon’s still pissed, and Rhaenys is way too young to be traded in compensation. On the other hand, it’s possible Tywin might back him if Rhaegar gives him Jaime back.