Well, sort of.
The water flow on a cold engine is "closed loop" basically through the heater core, block/heads etc. only. When the thermo opens up (about 200 degrees) the water can then leave the block/head go out to the radiator via the top hose, cool and then back into the system via the bottom hose and water pump. Depending on outside air temp and engine load, I suppose you could have a situation where the thermo never opens. Anyway, heat availability "should" be independent of thermo position.
With that said, a non opening thermostat should produce a hot engine and fluid back up/ overflow. The only thing is, I would expect heat availability in that situation. That said, you may have a combination of problems: a fair amount of air in the system and a stuck thermostat. I had to work at getting the air out of my system after recent work. A thermostat change is certainly no big deal. Be sure and get a new gasket with the thermostat.