the problem does happen, those stories of weird mainboard errors after cleaning the pc, well, sudden deaths of ethernet cards, sound cards, all that kind of problems surely are related o that esd situations
the only real problem is touch directly the chips , if you touch a choke or a capacitor, a connector or the soldiering points of those, is very likely that nothing will happen, but touch the chips directly is a bad idea, the material on them and the contacts will not have protection for sudden discharges of voltages coming from your hand and will end burned
cmos technology used on those chips is very sensible to esd, supposedly manufacturers have mad some improvements on the recent decades but cmos will always be sensible to esd
something that is not considered by most is that some houses can have improper ground installation and wiring, so the wristband could be doing anything