Why did the Tyrell’s not support Robert’s Rebellion, given that; they had surely been alienated from the Dornish-influenced court, would have been able to attack the Dornish rather than fight beside them, and that a rebel royal usurpation would have cemented their own questionable claim to Highgarden?

To begin with, the court wasn’t “Dornish-influenced” – Aerys was openly bigoted towards the Dornish and treated Elia as a virtual hostage. Rhaegar was a bit better, but his abduction of Lyanna would have been a sign that he didn’t particularly feel bound to his Martell kin.

Second, I don’t see how a royal rebellion would have shored up the Tyrell claim – if the Blackfyre Rebellion is any example, rebellions tend to be a time when lesser houses begin to get ideas about replacing their higherups.

Ultimately, I think the Tyrells sided with the crown because Mace thought he could get ahead by being a loyalist. Aerys was known for rewarding cronyism, and with the Lannisters out of the picture, the Tyrells would be the most prominent loyalist house around. Compare that to the Rebel Alliance, where the Tyrells would have been Johnny come latelys, not part of the original network of marriages and fosterages, and thus likely having to wait their turn behind the Starks, Baratheons, Arryns, and Tullys.