Zoro is a swordsman and he deals with his sword any danger he faces, the birdcage is a threat and Zoro must deal with it. What does he do, he goes there use his swords and cover it in haki BUT he doesn't try to cut it as his usual no he use his sword to push the birdcage. Lol you know how embaressing and degrading that is for any swordsman let alone someone who claims he wants to be the best of them all, that's like using a billions dollar note (if it existed) to wipe yourself after taking shit.
Zoro couldn't cut one single string of the birdcage without haki covering it, and Luffy destroyed a whole set of strings grouped together and covered in haki and going in a speed of a bullet. Clear and simple.
Don't bring this shit about not trying, we're not in Kindergarten where everything has to be explained in pictures (or panels in this case), Oda expect his readers to be smart enough to be able to put two and two together.
The birdcage exposed Zoro's weak haki comapred to Luffy's. Simple as that but of course Zoro fans will take whatever suit them and through away everything that expose Zoro.