Before you get out your tool kit, take a trip to harbor freight tools and buy yourself a thermal reading gun with the little "lazer" on it! Get ready to save yourself some time. I love them because when I making my rounds motoring, I can use the gun to tell me a bunch about my trusty "iron genny". Have some fun learning how to use it, lol, most belly buttons chime in at about 88 degrees!
The obvious, do you have water flow out your exhaust? That will tell you everything about the flow of water from that seacock you mentioned, through your raw water loop, and then overboard. If it is flowing, your not totally clogged BUT it may not be enough. That is critical.
You have an antifreeze loop. Good. How many hours on your engine? Probably in the 1000 hour range depending on your waters, it is time for an acid dip. However, that is a pain in the ass job.
So maybe a few IMHO ideas before you do the afore mentioned. You mentioned a nice stream at the seacock. Well where that tube next plugs should be your raw water strainer. Loosen the wingnuts and pull it out. Clean? Yes move on, no... clean it out and start her up. Bring her up to temperature and get out your gun. Or as you said bout five minutes or so. The seawater from your strainer to your raw water pump and up into the exchanger is not going to see big changes. Your antifreeze temp at several places is gonna tell you a bunch.
So in no particular order you need to check your thermo as mentioned. Shoot it with your gun, shoot the hose on both sides of it. It should be a 160 degree thermo so it should be pretty close when you shoot it. The coolant on either side of it should show some difference in temps. The hot side will be what just circulated through the engine and the cool side what just went through the exchanger.
Now find your way back to your mixing elbow. Its on the back of the engine. Its the exhaust pipe. So the hot exhaust is mixed with seawater in the mixing elbow to cool it so you dont melt your exhaust hose. Shoot it on both sides where the little water line from your exchanger screws into it. I forget the temp range but it should be a significant difference on either side. What happens over time to a mixing elbow is it basically gets heart disease and your seawater will not flow through it. But it being clogged is a different show.
This should give you an idea of where to start. IMHO, thermostats are to simple to fail and a pain to take out and test, but that is not to discount the expertise of ANY of the above posters. Some are better with tools than others.
So to sum up. Raw water seacock, raw water strainer, impeller, through exchanger housing, mixing elbow, water in exhaust.
Antifreeze loop. Temperature out of exchanger to engine , through engine and back to thermo.
Edit: Do you have a hot water heater that works from running your engine. Yep that is in the antifreeze circuit.
Damn I wish I had my engine to look at. Hopefully some of the guys will help make this easier to digest.
Cheers and good luck