I'm obviously not a fisherman...
If I like the flavors, is there any problem with cooking and eating a fish that hasn't been gutted?
Any particular species?
In other words, what are all the reasons people gut a fish before cooking?
Depending on your situation, you don't have to necessarily gut the fish, but in that case should cook it much longer than you otherwise would. Parasites are a concern, and the innards will make it harder for heat to propagate through the meat. Longer cooking times to ensure the insides are properly cooked mean a greater chance of overcooking the outer meat if you're not careful.
Another consideration in more and more places is pollution and mercury, and the concentration is going to be higher in some organs. I'm not sure of the scientific difference it might make, but for that reason, I don't want livers and intestines cooking next to the meat. I'm sure over long term exposure, you're getting it in the meat/flesh anyway, but if it is worth being concerned with, then taking care to avoid as much of what you can, doesn't hurt. Additionally, you don't know what the fish ate before you caught it that day. There might be something in the stomach/intestinal tract you wouldn't want cooking inside your meat...like plastic or other things.
Lastly, it's a good habit to not do it, from a survival perspective. Fishing is a great way to provide food in a survival situation, and you can move up the food chain, from a cricket or crumb of bread--or by crushing acorns and tossing them in a shallow pool full of minnows--to small fish, which can be eaten, but are often more useful as bait for bigger fish. From a minnow, a fisherman can work his way up to the largest fish, if he has bait and chum--and that's one reason why I don't cook or eat fish innards. I always fillet, and you then have tail, head, innards to use as chum or bait. Also, it provides bait for trapping. Raccoon, coyotes, wolves, bear and pretty much any predatory carnivore is drawn to the smell of fish guts. You will tend to do in a survival situation what you do out of habit, which is why you should get in the habit of seeing the fish innards as a potential health threat that in a survival situation, which might supply a parasite, toxin or illness that takes you out. And you should also see the guts as a resource that you should have on hand to increase your food supply.
Certainly, if you intend to store fish and not cook it immediately, gutting should be done. You don't want to try curing a fish with the guts intact, like some cultures might do with a pheasant. It does not work out near as nice.
Some small fish species such as the Smelt are eaten whole.
In some fish the appeal is in the flesh of the fish and are therefore gutted and deboned. Gutting can prevent some tainting of the flesh.
Like deer, the guts can deteriorate the flesh faster. One factor can be how fast you will refrigerate/ice your catch?
Another concern with specific types of fish is the content of the stomach (such as worms/parasites).