Can 12 gauge wire be inserted into 14 gauge switch connector sockets?
That is a side clamp switch, and it's your friend. Unfortunately, whoever wired it didn't do a friendly thing.
This image is a snippet from yours:

Where that red arrow points is called a &quot;backstab&quot;. It's quick and easy to install wires there, but, over time, the little piece of brass used as a spring clamp can wear out, leaving you with a loose connection. Loose connections lead to sparks and sparks lead to fires. Avoid backstabs at all cost.
Those green arrows point to the side clamps. You loosen the screw, strip your wire(s) bare, slip them into one or both of those little slots, then tighten down the screw. (Best bet is to use a torque screwdriver to be sure they're done properly.) Since these screws aren't under any stress, they are highly unlikely to ever back out without another application of a screwdriver and are, therefore, much more secure.
Just below where that wire enters the backstab, there should be a little rectangular slot. You can insert a small flat-blade screwdriver into that slot and wiggle the wire out. Screw it down under the clamp.
For the other wires on the other terminal, follow manassehkatz's suggestion to remove them both, pig-tail them with a 3rd piece of wire, then clamp the free end of that 3rd wire under the other screw. You can simply insert both wires into the two individual slots under the other screw and tighten them both down at the same time, but it's preferred to make a pig-tail and only have 1 connection to the switch. Either is code compliant.