Nonstarting Suburban

You need to make sure the battery has a full charge. If it is down, take it off and get it charged. My starter wiring diagram from MOD doesn't show a neutral switch, these databases are not always correct. The purple wire at the solenoid S terminal, should be hot with the key in the crank position. If it isn't, you will have to back track the circuit. I'd check the feed to and from the ignition switch first, before replacing switch.

You could use a remote starter button on the starter solenoid, just to see if starter will crank. There are safety issues, you need to make sure the tranny is in park or neutral and a functional e-brake is set.