You shouldn't let the same person shuffle and cut either. If you're dealing with a mechanic who could set a deck in the first place, it would be pretty trivial for him to perform a convincing false cut or to shuffle so that his cut will be the final step in setting the deck.The import thing is to do a "single overhand cut" directly onto the cover card (cut card.)
Then it makes little difference who cuts.
As far as ensuring fairness in game - the most important thing is that when a player is dealing, that dealer cannot deal their own cut. Never have the same person cut and deal, unless they are a dedicated professional and you know they're not cheating.