2000 Ford Expedition XLT 5.4, Possible Toast Heater Core?

Sounds odd. If the core was blocked, then hose water inserted in one side wouldn't come out the other. So I doubt its blocked. More than likely you have an air pocket, and that tells me your radiator cap is probably shot. On that year vehicle, I'm pretty sure there is a simple one hose radiator overflow reservoir. That reservoir is EXTREMELY important to the process of getting all the air bubbles out of the cooling system.

Here's how this works. When you drive your car, say for 30 minutes or more, the engine (and coolant) gets hot. Hot coolant can expand increasing the pressure inside the system to 13 to 16 psi. At this pressure, the boiling point of water goes from 212 deg F up to around 235 deg F (@13 psi) The radiator cools the car best when its full of really hot water, not full of really hot steam. The radiator cap maintains tight (and safe) control of that pressure.

Now here is the really awesome part of the process. When you shut the vehicle off (for the night) everything cools down. And as it cools down, that 13 psi turns into a vacuum, sucking coolant out of the overflow bottle into the radiator. This cool down procedure takes at least EIGHT HOURS. What makes this work in reverse is a little seal in the center of the radiator cap. See the photo.

Note, it will take about three or four days of this heat up, cool down procedure to purge ALL the air out of your coolant system. Note: I'm assuming that the hose from radiator to bottom of the overflow bottle is in good shape, no cracks, no leaks, etc.. I'll leave that inspection to you. Obviously when you start this process you will have to have the correct level of coolant in the overflow bottle.

It's also possible that you have a water pump problem. I've heard of old cars with worn out water pump impellers, but I think you'd see bad overheating on the vehicle temp gauge pretty quick.

Do check the radiator cap, to ensure all the seals are working correctly. I've seen many where the small seal disk doesn't retain well.

Let us know what you find. Good luck with it.

Oh, and one more thing... let's talk the direction of heater core flow. I'm not sure how your core is oriented. If the flow is down one side to the bottom and up the other side, inlet / outlet don't matter at all. (I strongly suspect this is what you have but I'm not positive.) If the core is oriented sideways, you always want the core to fill from the bottom. If you flow the coolant such that it fills the core up from the top, its possible to get a small air trapped zone in the core. This will cause a gurgle and you will definitely be able to hear it inside the vehicle.

edit:
Apparently this vehicle has a true coolant Degas bottle and not a simple overflow style coolant bottle. If that is the case, the whole radiator cap discussion is off base.

Again, can you blow air through the heater core tubes? hint. Remove both hoses from heater core. Take a three foot long piece of hose, hook it up on one side of the heater core. Blow coolant out of the core. With an empty core, you should easily be able to blow air thru the heater core coolant passages without even a hint of restriction. Blow from your mouth so you can easily gauge back pressure.

If core is good, you either have poor flow, or a blockage somewhere ELSE in the system. (Blockage) I've seen plenty of crazy things, including rubber parts stuck in the system. Anything is possible. (Low Flow) Thermostat stuck open, worn/degraded water pump impeller.

If the flow is blocked by oxidants then latent heat only lasts for seconds. Antifreeze ought to have antioxidants to minimize buildup of insulating materials.