What should I do with wires that are too large for back-stab terminals on a receptacles?
As long as you are going through the trouble to replace your receptacles, I have some recommendations:


Choose high quality models, especially those labeled "commercial" or "industrial" grade. They have tougher plastics that won't break as easily. I've often seen cheap receps break and lose bits of plastic, and they are left in service for decades.
Choose receps that have a contoured face. This makes it easier to get the plug in the right spot in the dark or behind furniture. Otherwise, it's tempting to use your finger to guide the plug in to the hole, which makes it easy to get shocked.
Get receps that have screwdown clamps. These are easier to install than wrapping the wire around a screw; more secure than stabbing in the back of those crappy receps; easier to remove than both.
Install tamper-resistant receptacles. These are code-required in many places, and they're a good idea.


As others have noted, it sounds like you're trying to insert 12ga wire in to a hole that only fits 14ga (for a 15A circuit). Don't use that feature; see #3.