TC is part of the ABS system. I believe on 2001-2004 cars it's a bundles package so you can't get ABS without TC and vice versa.
The traction control defaults to ON when you start the vehicle. The button is the means of turning it off. So if you key cycle and start, the ABS and TC light on the dash should go on and then turn off. At that point TC is armed. If you press the button, the LED light should come on indicating it's OFF.
It is common for tunes to delete the TC. It's not very good and was a good way to destroy an engine when you have a high HP combo get a little wheel spin at redline and the ECU pulls fuel. Not good. But with it tuned off, it should be defaulting to having the LED indicatior constantly lit indicating it's OFF constantly.
Unplugging the TC button itself won't disable TC.
So when you start the car, what warning lights come on and stay on?? DO you see the ABS light and TC light come on and go off? Do you have access to ForScan? Or a bidirectional OBD2 scanner?