I don't really understand why people are complaining about the Birdcage. Sure, you gotta be, like, crazy strong to actually manage to stop it, but it's not like we haven't seen stuff like this before. Remember Raigou, Enel's gigantic apocalyptic thunder cloud? Because I sure do. Unless a certain rubber man had appeared, that sh*t would've been unstoppable. He even made two of those things, which should be, I don't know, impossible to pull off or something.
Then there's the casualties of the Birdcage incident. Maybe I'm wrong here, but I think it'd be much more efficient for Doflamingo to get rid of the Dressrosians himself, instead of waiting for the Birdcage to slowly do its job. The cage isn't that strong of an attack, it just affects its surroundings greatly. So, in the end, it doesn't matter if it's Doflamingo or his Birdcage that does the job; if you're too weak, you're too weak. You're done. You can't win against Doflamingo. All the Birdcage does then is blocking your sorry ass to freedom. Basically, you gotta have the necessary strength to win against Doflamingo and his cage.
Also, it's not against the rules for a Devil Fruit user's technique to be stronger than the user himself. I mean, come on, the wake of Law's slash was capable of cutting apart Caesar's laboratory and the surrounding mountains. What's stopping him from one day doing the same thing to Dressrosa and all its buildings, while dismembering most of its population along with it? He could literally just stroll down the streets of Dressrosa and cut that sh*t in two with some swordsmanship, but I don't see people getting triggered over that. At least Doflamingo's Birdcage takes close to an hour or so to shrink completely.
I may as well talk a bit about Fujitora, as well. Obviously his greatest weapon isn't his psychical strength, so no one should expect him to be able to move the cage with a swing of his sword. However, using his usual gravity isn't exactly the smartest idea, either. Can he widen out the cage with his gravity? Probably, but in this situation? No. He'd have to push both sides of the cage, which obviously would affect the town and its people. They'd be moved along with the gravity. If he ended up only pushing, let's say, the north side of the Birdcage, the whole cage wouldn't widen out, but instead move north. The strings to the south would follow after and move across the remainders of the town.
"Why can't he go and fight Doflamingo himself, then?" you ask? If Zoro takes almost half an hour to reach the moving strings from the inner parts of the town, Fujitora, a blind man with a cane, from his current location, would probably take a bit longer than that to get to Doflamingo. And we all know Fujitora doesn't have that kind of time. The same applies to the Colosseum fighters, like Bartolomeo and Cavendish. They won't be able to reach Doflamingo before the cage shrinks. Neither do they know about Doflamingo's current condition, considering how the narrator's point of view isn't available to all. It's not like they'd just instinctively think Doflamingo's injured, either.
"Why didn't Fujitora fight Doflamingo before when he actually had time?" Because obviously he didn't know the Birdcage would start shrinking. Before that he was trying to do his job properly, believe it or not.