So the PC will not turn on - at all? Then it is probably the power-supply.
As mailpup noted; it could actually be
a lot of things - even the motherboard, but that is not very likely; I have come across several dozens of shortened motherboards (as an IT-tech.) and usually, more or less always, the PC responds to the power-button (it activates and simply sits at idle, without hard-drive "flutter", and with both power-LED and HDD-activity LED lit for about 40 sec.)
I would say that the PC-repair shop is probably correct in their diagnosis, provided that the PC makes no response to the power-switch at all.
As for the phenomenon of burn-outs from touching the power-switch, I actually experienced this personally last week, as my cat decided to sit on my external HDD
- and I can recall our physics teacher burning out the van der Graaf-generator after it had charged him electrostatically.
For now, you should try mailpup's suggestion.