Stability comes from the width of the kayak (hull shape plays a part as well). Most of the twin hull styles aren't actually wider than other styles and they don't really offer any real advantages. That's one reason they aren't real popular, they're also heavier and a little harder to control as well. The vast majority of kayak accidents are caused by plain old stupidity, like kayaking in flooded streams/rivers. Some of the cheaper sit in kayaks can tip if you're not careful, but the better designed ones and the SOP kayaks are near impossible to flip.
The foot controlled kayaks have their place, but won't work for everyone. They really shine for fishing because they leave your hands free. But like you mentioned, they really limit the type of water you can fish. I use my SOT in the saltwater flats of SW Florida and routinely go through water less than 6" deep.