Since I have been burned before giving advice on the wrong seceuity system in order for me to help you you must find out for sure either what version of a GM security system you have or what aftermarket system you have. The thing that steers me away from a GM system is that Passlock systems (the majority or) do not disable the crank system, they allow the car to crank,start, then die. Passkey systems (with a chip in key) disable crank. Now I don’t say it is impossible to have a starter disable Passlock system so you will have to look on the sticker on the inside of the glove box door (after contacting a GM parts man and asking what code to look for) and find out what system you have.
Just as likely your car suffers from acid damage from a previous leaking Delco battery or a normal fault that is inhibiting cranking. Any Passlock or Passkey or aftermarket system can be disabed.