The Tyrells were named by Aegon I as overlords of the Reach after all Gardeners died on the Field of Fire. This means that the Tyrells got their legitimacy as overlords of the Reach from the Targaryens, doesn’t it? Is this another possible explanation for their support of Aerys II during Robert’s Rebellion (as in: this guy secures our status, better support him)? After all, no one could know for sure that Robert would pardon them and confirm them in all lands and titles?