2001 chevy malibu - lights and radio turn on with key but when turning key to "ignition" won't get anything but a "click". No engine turn over. Tried to jump it - no. Direct to cables - no start

Clicking usually points to the starter motor.

I have the exact same car and had that exact problem!!!! Its the security system failing. I took it in and they wanted a ridicules amount of money to fix it. I did a ton of research and found a module that you can install yourself and it will bypass it and let the car start every time. I did mine 5 years ago and i havent had a problem since. Check it out they have great customer service and if for some reason it doesnt work they refund you 100% i hope this helps you out i know how frustrating it is
http://vatspasslockpasskeysecurityhelp.com/ Do It Yourself GM Security bypass