[RojoRocket]

I had the same problem a year or so ago, but the car usually started if I just continued to turn the ignition ON/OFF until it decided to start(usually within the first 10-20 tries). Again, never so much as a click when failing to start. Meantime the shop replaced the starter and ignition switch. As the problem continued to plague me, I was told the safety interlock (combination of foot on brake and shifter in Park, sensed by the Inhibitor switch) might be failing, and to try starting the car in "N". Sometimes it would, and sometimes not. The shop finally tried bypassing the inhibitor, and though the car now ran the risk of starting in gear if not paying attention, it would start, every time, first try.

Maybe I haven't so much answered your question, as made some suggestions on how you can move toward solving the problem by trial and error. Best of luck!

Glenn