I agree with ^
Okay, you have replaced the thermostat, we can eliminate that.
With the cold engine idling and the radiator full of water with the cap off,
after a several minutes, when the thermostat opens, you should see water start to come out the top of the radiator, that is if everything is working as designed.
As dr bowtie said; it could be calcium & rust has buildup inside your radiator and the water cannot get through it, down to the bottom to start the cycle over again.
As you know the Water Pump pulls the water from the bottom of the radiator,
then pushes the water through the bottom of the engine and out the top through the thermostat into the top hose and back in the radiator.
Have you replaced the Bottom Radiator Hose recently?
The reason I'm asking is there is a spiral spring inside the bottom hose to keep it from collapsing and if the old spring was not put in the new bottom hose, the water pump will not be able get enough water volume.
A Water Pump will always pump if it can get enough water to the impeller.
It may leak water out the weep hole, but it will still pump.
The impeller is turning when the engine is running.
Hope this helps!