Origin - Race: Both act as what your character was before their first bit of training and their genetics (or robotics if Power Armor/Cyborg/etc)
Power Themes - Power Source: Both act to define how your powers manifest. A "Might" theme (physical strength), for example, could have a variety of classes.
Role - Role: you still want each character to have a viable role in a party.
Classes: Combine a power theme and a role. Just like a Fighter is the Martial Defender.
Actually using this I think you can get what you are looking for with a slight tweak, keep races as a blank slot for compatibility purposes, and have Origin as a mandatory Background feature, so you'd effectively get two, except one give access to power sources. This even allows possibilities to change power sources if such an event were to be allowed (I can't think of any precedent in comics, but my knowledge is -FAR- from encyclopedic).
you just need to be a bit more flexible with the term 'class'. The origin and power source determine your pool of powers, and you swap roles on extended rest which all combine to create a variable 'class'. Same hero thematically, just a different set of powers for a different roll. Look to the F4 for a good example.
[sblock=Actual spoilers for people who don't read The Fantastic Four]Example the 1st: Richard is a Human Scientist with Rubber power source (REALLY specific I know, just an example, cosmic or mutated would be the proper source). He starts the day as a controler, focusing his power choices on powers that entangle and slow enemies. After the rest he swaps roles with Ben for Defender and focuses his power choices on powers that bolster his defenses and protect his allies. The entire time he is still Mr. Fantastic, but just with a trade in focus.
Example the 2nd: Susan starts the day as a leader, inspiring her friends and creating openings with her forcefields. After the extended rest she trades roles with Peter and becomes the striker focuses her powers on those that make her harder to detect and allows further setup for her allies. The entire time she is still the invisble woman, just her powers change focus
Example the 3rd: Peter starts the day as a striker, and does his spider thing, kicking people in the face via sky high web kicks. After the extended rest he starts using his powers to control the battlefield webbing his foes, and leting the other open up on the incapacitated bad guys. The entire time, he is still spiderman.
Example the 4th: Ben starts out as the defender. His powers focus on soaking damage. After the extended rest, he decides "IT'S CLOBBERIN TIME!" and goes striker... and, well, clobber's things. The entire time he's been The Thing, he just decided he was bored of getting hit.
Also for those of you who are saying "Hey wait, Spiderman isn't part of the Fantastic Four!" Your right, technically, but technically the F4 don't exist anymore[/sblock]
Its a system that has a HUGE amount of backing in the comic world, it just requires a massive amount of book keeping to keep track of all the powers. The powers themselves would have to be generalized templates that would require each individual super to flavor to match their own character. So in the end you would have Defender powers, Striker powers, Leader Powers, and Controller powers and once you combine a role with the power source, you have your 'class' as it were. In effect the class is a needless descriptor. This is kind of the point behind the 'Archetype' style of games, M&M and shadowrun are the only two I can think of. Games were your powers determine who you are, not vice versa (as in D&D). And having said this, I'd be interested in trying this if you ever end up running a game.