They are not only heroes, what about the villains with powers? It doens't matter when the power was obtained, people should hate and fear all "freaks" because they are all mutants by definition and many of them kill people.
In real life people complain about the media all the time, some people will believe, others will not, others will think that everything is a conspiracy from the government ect.. it is impossible that they could managed to convince the 100% of the population that the people born with powers is worse than the people who obtained by external factors.
The realistic conclusion is that people will not be able to tell the difference.
Like i said previously, i understand that with this type of argument sometimes we need to turn a blind eye and just accept that mutants are the only ones judged by their powers. The problem comes when Marvel push the mutant hate too far and the dissonance resurfaces.