I also have a 2.0 V6 merc. Engine has a thermostat located at the top of each head. Pee hole flows almost right away. However when running on the muffs only a small dribble comes out the pee hole but in the water I get a stream. I was taken aback at first when I got this engine because my old 90hp inline 6 merc had a strong pee stream on the muffs or water, this one has a bit of a weak stream even when in the water. I watched a few youtube videos and after some searching find this to be normal for this engine. When running on the muffs I have checked and the water coming out the prop does get warm like it should. I haven't installed them yet but I bought a temp gauge and water pressure gauge for some peace of mind.
Someone correct me if I'm wrong here but but the poppet valve plays into this as well. When the engine is cold and the thermostats closed the pressure built up by the water pump pushing against a closed thermostat forces the poppet valve to open, dumping the excess water directly into the exhaust cavity, so you immediately get water coming out the pee hole and prop hub. When the thermostat opens, pressure resistance drops, allowing the poppet to close. At higher rpm with a lot more water flow, even with the thermostats open pressure can be high enough to force the poppet open as well to dump excess pressure.