This is perhaps overly cynical; after all, many Vale Houses remained loyal to Aerys as well, as did a lot of the Riverlands, and some major players sat things out until after the Trident.
The people of Westeros take the idea of loyalty to their king, even a shitty king, very seriously. Even Stannis Baratheon talks about what a difficult choice it was; Aerys was burning people alive without trial and had declared Stannis’ brother an attainted traitor who he intended to also execute without trial, and Stannis still said:
Desperate for royal favor; Mace has wanted a daughter to be Queen and Viserys’ hand was open.
“If you only knew … that was a hard choosing. My blood or my liege. My brother or my king.“
(Davos IV, ASOS)
The Tyrell’s may have wanted Targaryen favor, but it is also entirely possible that like many, many others did, they simply decided “we’re loyal Targaryen men for three hundred years, we follow the Targaryens.”
That’s a fair point. There were plenty of people who supported Henry VI in the Wars of the Roses despite his manifest unfitness for the job because they just couldn’t get over the fact that he was a god-anointed sovereign whom they had sworn an oath of fealty to.
But Mace Tyrell does not strike me as a loyal anything. Hence his actions at Storm’s End.