Could be your brass gear on the crankshaft that runs the water pump and oil pump. The gear is pressed on the crank and if the tolerances weren't quite right between the two parts during assembly (crank OD on small side and/or gear ID on large size), the press fit wasn't as strong but would work fine when the engine is cold/cooler. When engine heats up the brass gear expands enough to spin on the crank under the load of turning the pumps.
This is semi-common with the Yamaha trippples, and doesn't always show up when the sled is newer. It would explain the heat exchangers getting nice and hot initially, then cooling off (as the pump stops working). Of course if you have gasket or bleeding issues as well, you need to get those resolved, but a good way to test the gear theory is to make sure you get the cooling system filled and bled, and no gasket issues, then run the sled keeping an eye on the temp of the running board exhangers. See if the exhangers cool down before the coolant blows out the overflow and starts to empty the system (in your desc above, the coolant bottle emptied before finding cool exchangers, and you may have had air in the system as well).
To fix would require a crank rebuild, letting the rebuilder know to make sure to fix the issue.
