"Fought for mutant rights" is a stretch. At the start he was a 2-dimensional villain fighting for mutants to rule over humans.

Direct quote: "The moment is at hand! All my months of preparation and planning shall pay off! The Human Race no longer deserves dominion over the planet Earth! The Day of the Mutants is upon us! The first phase of my plan shall be to show my power... to make homo sapiens bow to homo superior!" His plan involves invading a military base and taking control of "The mightiest rocket of all".

As for the status quo, the emergence of mutants changed the status quo. What Xavier is generally fighting for is that mutants have the same rights as normal humans. Magneto is fighting for mutants to dominate over humans. And, certain powerful interest in the government are fighting to either cage or exterminate mutants. The early years of the X-men comics are fights on one side against Magneto trying to do something with his group of super-powered mutants (literally called the Brotherhood of Evil Mutants), and on the other side fighting against powerful forces in the (human) government who see mutants as a threat and want to stop them before they get too powerful. Plus a bunch of fights against aliens, dimensional-hopping baddies, dinosaur-human hybrids from Antarctica, you know, the usual.

Eventually, Magneto is seemingly satisfied with mutants having their own country where they get to set their own laws and live separately from humans (backed up by the threat of force from the most powerful mutants), whereas Xavier wants humanity and mutants to live together in peace and harmony.