2000 Ford Expedition Heater

When the engine has warmed up, turn on the heater and feel both hoses going into the firewall under the hood and see if both hoses are hot. This will tell you if enough coolant is flowing through the heater core. Let me know what you find.

Here is a list of components that can cause low heat output:
A wrong temp rated thermostat will open too soon will cause low heat output (I know you just replaced it). A incorrect 50/50 coolant/water (more coolant than water) mix will cause a low heat issue. Temperature blend door half open or closed will cause a low heat condition. Air stuck in the heater core will cause a low heat issue. Restricted heater core will cause a low heat concern.

Does the engine get up to operating temperature? Does the radiator hoses feel the same as the heater hoses?
Did any debri come out of the heater core when you flushed it?
How did you check the temperature blend door? Did you see if the actuator is moving and/or remove the actuator and move the door by hand while the blower fan is on?

I think you have a restriction somewhere in the heater hose pipping. It's not the water pump or the engine would overheat or the pump would be leaking coolant. Ford used a valve on the heater hoses for a long time but I don't think yours has it. From what I've been able to find your heater hose has a flow restrictor to reduce flow of coolant and pressure to the heater code during high engine RPM (it reduces noise and steps down the pressure to the heater core). The items I mention earlier are typically what causes no heat.
When the engine has warmed up like after coming home from a trip, with the engine running feel all the heater hoses all around and feel for temperature differences specifically in areas where it is hot and then turns cooler (except the heater hoses going into the heater core and the other hose coming out. Because the heater core is a heat exchanger there will be some temperature difference but not dramatic). This will point to an area that is causing low to no coolant flow.
I feel it is a restriction because both the front and the auxiliary heater are not blowing hot heat.

It looks like you have a restriction at the t area. But before that, lets assume your water pump is not pumping, the engine would overheat due to no coolant circulation. Is doesn't seem like your engine is overheating. Lets assume your water pump is weak. Two things, if the impeller is the cause of reduced pumping it is usually due to the fins on the impeller breaking (plastic and or rusted metal) second, if too much clearance is created between the impeller and water pump housing adjacent to the impeller it would reduce pumping (if this is the case usually it is due to water pump bearing failure and coolant leaks).
When the engine is warm, have someone rev the engine in park up to 2,000rpm and hold it there then check the t fitting again to see if the temperature changes. If it does, good if it doesn't change temperature you have a restriction. Do this at the heater connections at the RR of the vehicle to feel if it is hot also. Have the blower on hot and blowing to see if the temperature of heat blowing out the vents is hotter with revving.
It doesn't sound like a water pump issue to me it sounds like a restriction or air trapped in the heater core.
I hope this helps, let me know.