Ida hasn't shown Zoro trying to cut them, so that's a plot armor right there. There's not excuse for it!
Dressrosa is miles across (exactly how many is unclear, but it seems to be at least several), and the Birdcage surrounds the whole thing. Assume that Zoro can cut through the Birdcage: what is actually accomplished?
If it doesn't compromise the structure of the birdcage, and the strings aren't under some kind of tension that causes them to snap unpredictably when cut, he's opened a tiny hole through which some few people could escape. Unless he's opening a hole like a mile long, it's not enough to significantly impact the situation. And in the worst case scenario, he could send the birdcage strings whipping through the interior of the cage… or bring the whole cage down on the island.
Honestly, it makes a lot of sense to me that, upon grasping the situation, Zoro and Fujitora aren't trying to destroy the cage.