France was a consitutional monarchy for a few years but Louis screwed it up by running
Not exactly.
Things were more complicated because in fact, the National Assembly and the king were at war with one another.
What differs from England is that the National Assembly had proclaimed that it represented the nation and that as such it was the sovereign. Never in England did you see the Parliament formally declare itself the sovereign instead of the king.
If you don't take this into account, you miss the point on how radical the french revolution was from the start.
So I would say that it's not by fleeing in 1791 that Louis XVI screwed the possibility of a french constitutional monarchy. He screwed it in the years before the revolution because he was incompetent as a ruler.
A better starting point would have been a success of Maupeou's reform.
You could also have Louis XV die young, just a few years after his son's birth. France would go through long years of regency and during these years evolve into a more parliamentary regime because of the need to finance wars.
For England, it's different. You need Charles I to win the civil war.