We go kayaking with the kids, and no we do not usr cold weather gear, wet suites and what not. But then, we do it in summer, on calm lakes or rivers, wearing floatation devices / life vests, everybody can swimm and we don't go out in bad weather (safety, but where is the fun getting wet and cold?).
That being said, you don't go out in an open kayak on rough water in cold weather without proper gear. And you do not advertise that this ok.
Where do you kayak? I scuba dive in warm waters with a rash guard and shorts or maybe a 3mm vest. My dive gear for Seattle is a drysuit with multiple layers inside. Please do not underestimate the thermal conductivity of water. The bad news is that cold also slows your decision making leading to making more stupid decisions.
In this particular case, they are showing people kayaking with shirts, shorts and a PFD around San Juan islands - notorious currents, water channels through a narrow inlet and water is really cold year around. If they had shown the same people with same gear in a Caribbean location, it would have been ok. And the other person kayaking in alpine lakes with a thick jacket is even more crazy. When (not if) you fall in, it gets water logged and pulls you down.
That being said, you don't go out in an open kayak on rough water in cold weather without proper gear. And you do not advertise that this ok.