It should look like this. When the belt is off, it is a good opportunity the check that all the components on the belt route turn okay and they is no play in bearings, apart from Crankshaft.
Squealing is caused by the belt slipping or a pulley not turning properly. Does the alternator spin feely as charging seems like it might be an issue here. For belt slippage you can spray with a belt conditioner which helps grip.
https://beltposter.blogspot.com/2020/09/2007-saturn-aura-35-serpentine-belt_30.html
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Battery could be shorting out itself between cells. When it heats up it shorts out when it cool down it doesnt. This is one reason i like interstate batteries. They are made of all new material. Most brands are rebuilt batteries.
How hot does the engine get in degrees F? Does the radiator fan work? You may need to check coolant flow through the radiator and heater code? With a warmed up engine, are the heater hoses hot? If the blower motor works ok, just not warm, check temp actuator, probably under the dash.
Any applicable trouble codes?
There is no on-board programming procedure for this vehicle. This vehicle requires special equipment be connected to the vehicle to program the key-less remote.
That may be just one of many things now wrong with your car.
Please check that every other electrical component still works, make sure that all ignition lights and gauges still operate.......if the only thing wrong with your car is the radio, you got off easy.
The radio is probably toast.
