2002 Pontiac Aztek Engine Overheating

Something was lost in translation. A "whole cooling system" is never replaced. A part is replaced if it is worn or leaking, and the coolant can be flushed and replaced, typically every two years since that's how long it takes for the additives in it to wear out.

Since you already made the dandy observation that the heater hoses are cold, it suggests coolant isn't flowing to the radiator either. Feel the upper radiator hose when the engine is hot. If that hose is cool too, first check if there's a bleeder screw on the thermostat housing. Open that to expel any air. If there's no bleeder screw, look for a sensor or other plug that can be removed temporarily. If you unscrew a sensor, do that with the ignition switch off so you don't set any diagnostic fault codes that can be misleading later.

If you continue to find air in the system or the overheating continues, have the cooling system checked for a leaking cylinder head gasket. Combustion air sneaking into the cooling system can pool under the thermostat and prevent it from opening. Thermostats have to be hit with hot liquid to open. They won't open in response to hot air.

The chemical test involves drawing air from the radiator through a glass cylinder with two chambers partially-filled with a special dark blue liquid. If combustion gases are present, the liquid will turn bright yellow.