The Tyrells may have the largest military force, but they're not gonna cast it away on a lark. I think the Tyrells, as much or more than the other houses, know how to go with the flow. Mace, allied with Dorne and possibly the Greyjoys, could have fought Robert and Ned for another year or so. Mace also could have found someone to father a child on Margaery, call the child Renly's heir, and have continued the crazy war of the Five Kings. Neither time did Mace push his luck. As soon as an easy victory eluded him, he quickly made peace and got on the winning side.
Mace and Olenna remind me of Robert the Bruce's father in Braveheart... always scheming, never commiting, ever vigilant for opportunities. If they can get one of Mace's children to be one of the heads, then they'll do it. If Dany refuses them, then they'll bow the knee and wait for another chance.
The Tyrells are aware of a number of facts. First, the Martells are their mortal enemies. The Iron Throne counts on this enmity as part of it's checks and balances to provide stability. Neither family will let the other get too powerful.
Second, the high lords of the Andals, i.e. the Lannisters and the Arryns and the ancient families that serve them view the Tyrells as upstarts. The Tullys and the Tyrells greatly benefited from Aegon the Conqueror. He promoted both of them to overlord status. If someone like Tywin Lannister was the real power in the realm for very long, then the Tyrells might find themselves replaced by another upstart... this one grateful to the current regime. Also, because they are viewed as upstarts the Tyrells are probably less able to lead a victorious coalition than the Lannisters and Arryns.
Third, they know they are powerful. They may be newcomers, they may have archenemies, but they have great military and economic influence. Only by being completely foolish can they be omitted from any alliance that will rule Westeros. All of their competitors must account for Highgarden. So if Mace and Olenna are open to offers and circumstances, they are likely to gain more power.