They're completely correct. As a Floridian, I can tell you that I've personally swum in waters infested by sharks and had no problems. I got bumped a few times, but they never attacked. There were at least a dozen sharks (mostly hammerheads) and none of them bothered with us. It was somewhat invigorating if not unnerving, but if you don't present yourself as prey (flailing or swimming on the surface or looking like food from below), they won't bug you.
We used to walk the inlet from north Clearwater beach to Caladesi Island during low tide when we were kids, and we'd see hammerheads, nurse, and bull sharks all the time. They'd circle us and go between our legs, but the water was only knee-high. We knew the first two were relatively harmless, but every Floridian has heard stories about the bull shark. If I see the hammer-shaped head, I don't worry, but a bull shark is something to avoid. Will they attack a swimmer? Hard to say, but the more prey-like you look, the more likely they'll come in for a test bite.
So how do you swim, if not on the surface?
I have talked myself right out of the water more than once after feeling something bump my leg.
Yes, it is irrational, but the idea of sharks near my legs scare the heck outta me.