It should be sufficient to discharge any static you have accumulated by touching something metal or grounded. e.g. the chassis of the PC, or metal housing of the PSU while it is still connected to an outlet (assuming it has a real earth ground). I have a couple of those wrist straps, never used them (well I did try it on once but it was annoying) and have never zapped anything. I think in certain environments it could be a good idea if you are surrounded by static potential, where you could actually build-up static charge again while working on a computer even though you discharged immediately prior to opening the PC. But I've never had a problem.
Working inside the old CRT monitors and televisions, however, is another thing. There, the grounding strap is to protect YOU not the equipment.