It would have had a shot, but there are still huge tensions in the political system that I think resolution would be extremely difficult indeed. For one thing, Aerys’ loyalists are going to be heavily resistant to the new regime and there are going to be no few people who will think of Rhaegar as a usurper and (if Aerys is put out of sight) there will be rumors that he’s a kinslayer too.
For another, I think the Southron Ambitions bloc isn’t going to follow Rhaegar’s lead down the line. They clearly had their own agenda and there are going to be moments where that conflicts with Rhaegar’s agenda.
And finally, there’s some powerful factions that are out-of-pocket – the Lannisters, Tyrells, Greyjoys, etc. They’re going to try to exploit the uncertainty and division for their own interests.