How would the Reach’s role in post-Conquest Westerosi politics be different had the Gardener line not died out at the Field of Fire?

The main difference has to do with the relative unity of the Reach, given the relative weakness of the Tyrells’ position – so you wouldn’t see the Reach splitting during the Dance or the Blackfyre Rebellions. 

But also I imagine you’d see a different relationship with the Iron Throne – whereas the Tyrells have always sought royal favor b/c of their desire for a royal marriage, the Gardeners would have been more like the Starks or the Lannisters or the Arryns.