Analog microphone audio is one wire (and ground) having the AC signal of the audio, superimposed on a DC signal powering microphone capsula. The simplest way of making a killswitch is to either 1) short the signal to ground or 2) cut the signal between mic capsula and the soundcard input. Done with just a switch, both of these will impact the AC component as well as the DC component, and the DC offset change that causes the pop.
And yes, there are many ways to avoid this problem. I think adding a resistor and capacitor to form a high-pass filter for the shorting option would work fine. If there is already a PCB for the switch, adding these two components would cost practically nothing.
The "pop-less" microphone switch is generally a series R-C pair, where the R is, say, 1 MOhm, and the C a few µF. The switch shorts the R out; the R charges the capacitor to the bias level when unmuted, and so shorting the R produces very little pop. The capacitor then shorts the AC audio component.
XLR switches are easier, just short hot and cold, done. Works with all microphones and doesn't produce a pop, because XLR uses phantom power instead of T-power.
And yes, there are many ways to avoid this problem. I think adding a resistor and capacitor to form a high-pass filter for the shorting option would work fine. If there is already a PCB for the switch, adding these two components would cost practically nothing.