I've done alot of kayak fishing on the bigger lakes around Saskatchewan Canada, i havent had a problem with tipping yet and i hope to never haha. the one thing i find to be a pain and only pain with the kayak is landing your catch, i mainly catch Northern Pike and they fight like nuts, sometimes i gotta drag them to the top with one hand on rod and other on the net, biggest i've caught was a 10pound northern pike. my favorite part of kayak fishing is, you can get some good speed on them, they are easy to load and transport, and you can fish in very shallow water, and creeks thats boats cant get to.
few tips: 1 Only bring a small tackle holder with your favorite lures. makes it easy to find your lures and not loose all of your tackle box if you tip.
2 get a small trout net to keep on the kayak cause it can be sometimes hard to grab fish with your hands (sometimes due to size, or stuborness)