They had PR to try and fix their already horrible public perception which was built off of several years worth of being hated and looked down upon. Cyclops hiring a PR firm for a few months wasnt going to make a dent in that
It doesnt help when you have politicians with agendas that rally up people and spread their hate. It shouldnt be too hard to buy as this literally happens all the time in real life. From the first issue, Magneto's attack on Cape Citadel gave them ammunition to spread anti-mutant proaganda and for any progress the X-men have made to detract from that, tons of evil mutants have set them back. They've made it a habit of taking in those same villians that the public have seen be terrorists and murders and its no surprise that people are confused and hate them. It would be like the US government taking in Osama Bin Laden and trooping around with him like he was our own
The Avengers are backed by the government and even when questionable things go wrong for them, they tend to be quick to clean it up and have an amazing PR team to keep public perception high. They have the government helping them out which comes with privileges that the X-men arent afforded