Out of all my experience..PSU are least effected by static electric charge.
I never used a antistatic wrist strap, i thought it as annoying, however 1 thing you must do is that you wash your hands with water for water is a conductor of body electricity the other is that i always touch the painted side of my case to achieve perfect grounding
The meaning of grounding is that you achieve the same charge or equal charge as the Electronic equipment itself.So most often than not it is not the case of anti static strap or not but the equal body charge with the charge of hardware.
That is the means of grounding.
Washing Hands - Yes that should drain the charge from your body as tap water, unlike distilled water, is a conductor. BUT while walking back to your system, you just built that charge back up.
Touching Painted surface. Body (conductor + paint (Insulator) + metal under paint (conductor) = a Capacitor, not a path to ground to drain charge. Use to demonstrate this principle by using a screwdriver and brining it close to 3500 V DC. Again you as conductor, plastic handle (insulator) and metal blade. You would see a arc as you charge this "Capacitor", then remove the screwdriver and bring your finger (other hand) close to the blade and you see a arc as you discharge that "capacitor"
Touching a painted surface only works if your voltage is high enough to brake down the paint. Need to touch a UNPAINTED, bare metal surface that is tied to earth ground.
Myself, I always wear a ESD strap when handling MB, CPUs, memory modules, and circuit cards. The PSU is fine as it has a rather large faraday cage surrounding the components.
Google ESD to understand what ESD is the the damage it can cause.