Anyone know if I can repair just this worklight switch/circuit
(replace the wires?), or will I probably have to replace the whole
control panel (I've seen it for ~$50 online--ouch)? At least if I do
have to replace it, it looks like it just pops in and out, and the
connectors do the same. Any advice on this is appreciated; thanks.

Does your microwave normally beep when any buttons are pressed? If so, and
the light is the only one which doesn't beep, you'll know it's the membrane
assembly.

My guess is the triac or whatever solid state switch - or maybe even
mechanical relay - is bad. Listen closely for a click inside the unit when
you press it. Could be it has a relay with carbon loaded contacts.