How do you think the history of Westeros would have changed if the Gardener line had survived the Field of Fire?

Good question!

Well, the Gardeners bend the knee like the Lannisters, which means the Tyrells remain the savvy stewards they were before. This in turn means that the Reach is going to be a lot more politically unified in future conflicts – the Revolt of the Faithful, the Dance of the Dragons, the Blackfyre Rebellions, etc.