If the 2001 is like the 2000 Alero this may help.
""Pry up the plastic molding from around the radio/heater A/C controller and from around the gear shifter. Detach the heater A/C module from the molding by removing the four 7mm hex bolts from the backside. Take out the three 7mm hex bolts to loosen the radio. Put the key in the ignition and turn to "on" position, apply brake, and shift into "drive" or lower gear. Turn the heater A/C module to the left, sideways orientation, and pull out the radio. Disconnect main wiring harness from back center of radio if there is not enough slack to take the radio out past the plastic molding. Leave the antenna connected. Look inside and to the left at the back of the ignition cylinder assembly. Locate the thin yellow wire, which is bundled alongside the thin white and the thin black wire.""
But again, if you mess with that wire, as in cut it and wire in resistors that match your key pellet, then that fix is only for a bad ignition switch.
The TDM that would be the culprit if you rule out an intermittent short in the ignition switch wiring.
I tried to find more info on exactly where the TDM is, but all I could find were suggestions that it is where I mentioned, behind the IP Panel on the passenger side, you pretty much have to turn upside down to get down in there to get to it, at least I did.
I think mine was somewhat near the Body Control Module..
I am a little tired atm so my memory is not as sharp as it should be, badly in need of a long nap!
This is what your Theft Deterrent Module looks like.
Here is what the wiring to the switch that I am talking about looks like.
I still would get a multimeter, and when the car won't start, then set the multimeter to ohms, and find that connection coming from the ignition switch, the connector in the switch above is what I am talking about.
With the key in measure the ohms from the 2 wires contacts via back probing them, then measure the resistance from the contact on both sides of the keys pellet, then compare resistance values, they should be really close to the same.
Values should be one of the listed values in this pic.