Standing in any yak in moving water will eventually catch up to you.
Those native watercraft are unbelievably stable. I've taken it down class II-III rapids standing up. Honestly, for me the risk of hitting a rock and falling out is outweighed by the fact that I can see the rocks so much better when I'm standing. If you're in fast water, you have to manage your speed carefully but you can definitely do it.
The other, potentially safer strategy is that you can sit down anytime you like. There is no rule that you have to remain standing just because it is possible to stand. The seats are comfortable.
The biggest risk is when you get big a standing wave that comes over the side of the boat. That happens pretty much every time I go down class III, at least to some extent. I usually have to stop and bail the boat out. Sitting/standing won't change anything there. The choice for that situation is that you bail the boat, buy a kayak, or buy those inserts that whitewater canoeists use.
Avoid class III whitewater is probably not a bad policy if your goal is to catch fish. You cannot simultaneously manage both.
The stand up type yaks are normally a lot heavier and that means much more of a chore to get on/off the roof. More of a chore to drag through the brush to the spot you want to launch. This will eventually get old and you will use it less often.
This may be true. What I really wanted was a canoe that I could stand in, which is basically what those native watercraft are. Mine weighs about 75 lbs. I worried early on that the boat would be too much trouble and I'd find myself wade fishing, but it's actually gone the other way. I use it a lot, and its become my preferred way to fish for trout and bass.
My dad has one of those hobie pedal kayaks, which I forgot to mention in my first post. Those are nice, but you do have to be careful with the flappers on streams with low water. It is ideal on a lake since you can cruise the shoreline with your hands free for casting. Switching between paddles and fly rods is definitely not a skill that people are born knowing how to do.