Switch designed to take #14 only

Don't use the back stabs, they are often the cause of failures. Use the screws on the side instead. If the switch doesn't have screws, buy an other one that does.

And NO, you can't use 14ga on a 20 amp circuit anywhere. Just get a high quality switch, it's your least expensive and best option.

Back to the store it goes.

Switches which only take backstab connections are the cheapest of the builder/slumlord grade cheap.

Backstab connection, in general, are bad news - while legal, they are widely known to be unreliable. They tend to knock out half your circuit and send you on a frustrating "bug hunt" - hang out here for a week and you'll see 2-3 such questions pop up.

Further, backstab connections can only be used once because pulling the wire out fatigues the spring. I have a hunch this is a used one, and into the trash that goes!

New builder-grade switches are about 75 cents and have side screws. New spec-grade switches are $3 and allow screw-to-clamp (2 wires under each screw; no backstab) which is safe to reuse.

No part of a 20A circuit can be #14 wire.

None. You simply need to use appropriate hardware. That ain't it.

You do not need to use the expensive 20A-rated switches, unless the load you are switching is actually that big :)