If you hear a click from the solenoid and you can see a change on the ammeter, then the problem is not in the ignition switch. It's in the starter.
I had a similar problem where my starter was a rebuilt unit installed by my previous owner. Intermittent operation where it made a loud "CLICK", but wouldn't engage the starter motor. Found that the Bendix unit on the stater had been assembled incorrectly by the rebuild company. Been a number of years (and I didn't take any pics), but fading memory says they had installed a spring incorrectly. On the wrong side of something.
After seeing the problem, I was surprised that it ever worked at all, even intermittently. Anyway, took it apart, relocated the spring, and it's been flawless since.