No Start condition

I had the trans replaced a few years ago, after they did the work, I pulled away to go get gas. After filling up, it wouldn't start. No click from the starter, fuel pump made noise, all the lights worked. They installed a push button, said the problem was my ignition. Two years later, I'm faced with the same problem. My starter works, you can jump with screwdriver. but won't work with the button they put in. The security light flashes when the door is open and closed, but once the key is in, it goes out. The morons put a wire to the 12v wire going to the starter, and another wire to the S terminal on the solenoid. The wires are the same color..... Normally I would put key in, wait for the security light to go out, hit the button and away I go. The relay is good. Could my ignition have went completely out? Its a 2000 Bonneville with 3800 non supercharged.

Basic checks if the sec. lite is out, go to the solenoid on the starter, then have some one depress this button ( ridiculous repair on a modern car!), check for 12v at the solenoid when the button is depressed if none that button or circuit is bad, if so it is the starter. BTW usually when this occurs after a trans job the prndl sensor/switch on the trans is bad or misfitted or the connector is loose.

Thank you for responding. The starter is good. The wires to the push button are good and intact. I started pulling out the ignition switch, but I'm having trouble with the single wire that plugs into the back of it. Its a very thick wire. I pushed in the retaining slot on it, and it comes out about half an inch, but won't budge any further. The transmission that the guy put in wasn't the correct one. Its cast on top for a neutral safety switch. I was told from a different shop that did a road test for me that its the wrong one. It doesn't have a clue what gear its in. That the NSS is inside the trans for my car which is why I couldn't find the plug in connector for it. Once I get the wire/cable off the back of the ignition, I'm putting in a new one because it had been causing me problems. What does the cable/wire go to? Its held me up for 2 hours without coming off.