Stability, like anything, is relative. Did you ever learn how to ride a bike? It was pretty tough at first. Everything we learn is. Take walking. We weren't very good but now we walk easily. Kayak fishing is the same. Obviously its a great way to fish as so many of us are doing it now. Its not mass hipnosis. If you very concerned about stability get a very stable kayak. You'll sacrifice a few other things but go for the attribute you want the most.
In well over 100 days kayak fishing I've only dumbed a kayak twice unintentionally. Once was a surf landing with a very hard break that was impossible to handle in a yak without a rudder or fin. Also I was in a kayak I had never landed in the surf before. The other was very recently. A couple guys were motoring by me slowing asking questions about the yak behind me. As I they went from port to starboard I kept talking with them and turning and violated the rule of stability and had both shoulders on one side of the yak. BAM, I was in the water. So I just climbed back on, no big deal. Its easy.
A pontoon boat is not a good choice for fishing saltwater in this area. IT is so limited as a fishing vessel compared to a kayak. Its amazing what and where a kayak will do and go when properly chosen, equipped and piloted by an experienced operator.