the cooling system raw water pump (impeller) circulates water all the time. depending on the temperature of the engine, this water goes to different places. the thermostat controls where the water goes: when the engine is started and running cold, all of the water sucked up by the raw water pump goes right into the exhaust. this is to effectively 'overheat' the engine to get it to come up to a reasonable operating temperature. once the engine warms up because the closed thermostat is making the engine get wamer, the thermostat opens and allows some cold water to pass through the cooling passages, or water jacket, of the block. this cools off the engine to it's happy temperature or even a bit colder, the tstat closes again, warms the engine up, opens again, etc etc until engine and cooling water temperature reach sssf.
whats left always goes into the 'mixing elbow' or exhaust to keep it cool enough from starting a fire...
the danger with winterizing - you could theoretically run anti freeze though a warm engine and get anti freeze into the water passages of the block. if things are not warm enough, you won't get any antifreeze into where it counts. definitely safer to pull the thermostat, then you know you have antifreeze through the whole block. i've seen (not on my boat) frost plugs sitting in the bilge in the spring, luckily they didn't end up with a cracked block.....