Heater blowing cold air

What is your question?

So, here's what I've done: (1) checked coolant level, (2) confirmed blend door is working perfectly, (3) confirmed that both inlet/output tubes to the heater core on the firewall are hot, (4) torn apart the dash and confirmed that the heater core itself is hot to the touch, and (5) reassembled everything to get the same room-temperature air blowing when the temp is set to full hot. I have front and rear heat, and both are functioning the same way. I was assuming the heater core was plugged, and that I'd just replace that, but (a) the inlet/outlet, and the heater core itself is hot to the touch, and (b) I've read a bunch of posts from people who replaced the heater core and still had problems. So now, I'm asking myself, what is it really that's going on? It seems like the heater core is hot, so the heat gives me an initial blast of warm air, but as soon as the air flows over the heater core, it seems like things cool down quickly, so all I get is the room-temperature airflow. Any thoughts? I'd love to disconnect the hoses feeding the heater core, but they have some sort of clip connect tucked up under the firewall that I can't see and figure out. Any info on how to get those disconnected, too (without cutting), much appreciated.