Because he couldn't cut the cage, that's why. And Birdcage seems to have been on another level as it cut Fujitora's meteor, if I recall correctly, and it was pushing buildings to the epicenter, while Luffy was able to break out of Doflamingo's binds. Birdcage seems to be superior.
No, Birdcage is made from strings of the Black Knight which fodder Corrida combatants have cut through.
The meteor is just stone or whatever, not a hard material. Fujitora who wanted to save the citizens too apparently couldnt cut it either, not that he tried either.
The Birdcage simply had plot armor to keep the suspense until Doffy is completely defeated. Fujitora saving citizens by cutting the Birdcage doesnt defeat Doffy himself.
The very fact that it never occurred to anyone to even try cutting it speaks for itself. It is Zoro who we are talking about, the man who hammered on Mr1 until he has gone through. The man with superior Haki, strength, weapons than Doflamingo. And the strings which Abdullah and Jeet have cut through.
All facts from the manga. And again, Birdcage is irrelevant in beating Doflamingo.