Yam should pump water all the time.
Dont mess about with Coke (or vinegar), get some Rydlyme to flush the engine and clean the thermostat. As mentioned above you can run the motor without the stat but its not great for the performance especially in our cooler temperatures. Some engines dont work well with the stat removed, it can mess up the water flow and reduce cooling (the Yamaha 6/8c is like this).
Strimmer line is great for working your way round the cooling system. Great advice about basic service items above, I would also change the gearbox oil and gaskets at the same time.
Rydlyme Marine Descaler