Okay, so it's not the starter solenoid ("magnetic switch") that's clicking.
At the back of the starter solenoid there's three connections. Two are heavy bolted connectors. One of these has the cable coming from the battery; the other has a wire going down into the starter motor itself.
The third is a small, single wire with a spade-type push-on connector. This is the one that provides power to the solenoid when the key is at the Start position. If this hasn't been done, I suggest measuring the voltage at that connection; when the key is turned to Start there should be approx 12 V.
If there's approx 12 V there, then the starter solenoid is bad. (It isn't activating to connect the heavy battery cable through its internal contacts to the starter motor winding.)
If there isn't approx 12 V, then there's a problem in the starter system. The inhibitor switch and the starter relay are involved, but also the security system through the body integrated unit and security relay. I've attached a wiring diagram which I believe applies to your car.
The aftermarket remote start system could be a factor as well. I have no experience with these, but from what I've read here, they aren't just "plug and play"; installation involves some wiring changes. If this is the case, then they can't just be disconnected to take them out of the system. Rather, once they're disconnected, the wiring of the car has to be returned to the original state. If that's the case (and I recognize I might be wrong), then troubleshooting might require more than just disconnecting the remote start module.