It didn't really exist until Len Wein introduced the All New, All-Different team in the mid-70s. Prior to that, they were publicly acknowledged heroes and adored by civilians, had a direct line to Washington with FBI and CIA contacts and villains like Spider-Man's Sam Bullitt furthered their political careers by pretending to be affiliated with the X-Men.
There were a few instances when the public turned on them, but no more than they did The Avengers & The FF, and a hell of a lot less than Spider-Man. Trask and his son were portrayed as gibbering nutjobs who, even after gaining support, were still opposed by most of the populace. The few cases of anti-mutant sentiment came from one-off villains who hated superheroes in general, people who caught the X-Men in civilian guise (and thought they were supervillains in hiding) or torch-wielding superstitious European peasants who thought mutants were witches/demons. Even the famous "We paid out fare like everyone else" panel was just a joke about how silly superhero costumes would look irl; Daredevil and the FF did the same gag all the time.
The whole "Oppression story/Metaphor for civil rights" is pure retcon bullshit.