What about a tier/hierarchy-class system:
One "signature" power (Charge, Tech Armor, etc.)
Two "main" powers (Nova, Overload, etc.)
Three "extra" powers (ammo, grenades, Pull, Lift, etc.)
One "passive" (one that emphasizes weapon damage, one that emphasizes power damage, etc.)
Or going further, having "basic" powers a bit like ME1, and as you play you find other powers, like omni-tool schematics or biotic implants and such.
Like this sample "Vanguard":
You start with...
N7 Training (passive)
Pistols (weapon)
Biotic Rush (a mild version of Charge, think lunging melee punch at high speed)
First mission, you investigate a comms blackout on the Ark, and you find it's being hacked by pirates, you beat them, save the day, and find "Omni-tool Flash-Forge Data: Incinerate" on one of their omni-tools.
Congratulations, you can now learn Incinerate or teach it to your squadmates.
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Perhaps doing away with set classes entirely, i.e. rather than picking one and sticking with it through the whole game, start basic and then "gain" a class of your choosing, with the option to switch whenever you're on your ship. With the caveat that if you are a level 20 Vanguard and want to try Sentinel, you start as a level 1 Sentinel, and now you have to level up as a Sentinel. You don't lose your Vanguard levels, but that could introduce some interesting gameplay options; do you stick with one super-powerful class, or multiple, less powerful classes?
Going along with that, it wouldn't be a simple "press 'X' to switch", there would be a cost, similar to the power reset in the Normandy's medbay, relevant to the class you are compared to what you wish to change to, and the larger the difference between the two, the greater the cost.
As in, going from Soldier to Vanguard would be comparatively cheaper than going from Soldier to full-on Adept.
I think I've mentioned this in a few threads, but there's so many of them now it's difficult to find them all.
In any event, there's my two cents on the matter.