Yes you would. The thermostat just controls the hot water admission to
the radiator. The heater is direct from the hot side of the engine. If
the flow to the rad is blocked it will boil up and probably eject water
from where the hoses leak easiest. If this is low down then the leak
will be cold, that's why I suggest the bottom hose needs checking.
Fairly obviously you aren't getting any flow through the radiator or
everything would get warm