2001 silverado, 5.3L. heater control stays hot,

Yes it would overheat without circulation, but the coolant doesn't circulate through the overflow tank. That reservoir only gets coolant into it when the coolant gets hot and expands and pushes the radiator cap seal up to let coolant into the reservoir. When the vehicle cools down, the coolant contracts and the lower level in the radiator will siphon the coolant out of the reservoir back into the radiator. Coolant circulation is from the engine to the radiator and back into the engine...and also through the heater core.
You may have an air block in the heater core. Check by taking off the outlet hose at the firewall or at the engine and see if you have a slow steady stream of coolant, with the engine running and heater controls on high heat. If it was an air block, a steady stream should quickly start. If you get little or no flow, the core may be plugged and need flushing backward or replaced.

If it has dual zone temp. control it is most likely the heater control panel.

My truck had the same issue and from I have read, it is pretty common. I just felt the heater hoses under the hood. If there is one hot and one cold, with the engine running and the heater set to maximum, rapidly squeeze and release the hot hose for about fifteen seconds.I just did this to my truck and it "Burped" the heater core. I got instant LAVA HOT air!! I hope this helps.

Do you have a hot water valve out by or on the engine usually you can tell by following the heater hoses the water valve will be in line or be coming off of the engine intake manifold, it is vacuum operated older valves can stick, try that first or if you have a vacuum gauge check to see if the vacuum bleeds off you can also manually apply vacuum and feel what the heater temperature does.

Yes, you probably have air in the system. Park uphill and run the engine with the heater on to get air out. Refill radiator. If air comes back or continues to bubble up in radiator, you may have a leaky head gasket.

Your heater core is clogged, leaking, and/or going bad. If it was anything else in the cooling/heating system, you would not have the antifreeze odor. The heater core is located under the glove box. I hope this helps and good luck.
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no this is not a thermo issue the heater controls are bad and need to be replaced ive seen this alot in general motor trucks and cars.thanks and good luck

There are a couple of things you can check, or have checked. Under the dash, behind or around the ac duct work, there is vacuum operated actuator valves, which open and close different doors to direct heat or cool air,one of them might be stuck in the heat position.But the first thing to check, if the truck is equiped with it, is the heater control valve, it is under hood passenger side connected to one of the heater hoses going into firewall.Move the lever in opposit position, if you get cool air,thats the problem.If it's vacuum operated,check for leak,if no leak,replace valve.

I would check your heater control valve, that, or the core itself being bad, are the two most common problems I've had with Chevys.

Sounds like the blend door in the ducting has failed, this door is what controls the temp of the outlet air by blending heat and outside air or AC cooled air.