[john70t]

The car can be push started consistently, so fuel and ignition module/relays can be ruled out along with the "on" key circuit.

The clutch switch and the "start" circuit of the ignition switch are suspect, but what about the starter itself?
-The ignition switch in the "start" position sends power to the solenoid. That connector could be loose at the solenoid.
-Starter solenoid benzels which stick sometimes can be "fixed" with some light hammer-taps.

Here's how it works:
The 12V from the ign switch(control circuit) first pops the starter gear out to engage the flywheel with a "click", which results in the internal switch(load circuit) drawing heavy amps through the thick red wire from the battery to spin the motor.
It could just be a sticking benzel.

Check if there's 12V supplied to the starter solenoid (small wire) when cranking, and all connection are tight and non-corroded.
Hot conditions are prime for electronics failure, but it could be something simple like the starter, bad clutch switch, or bad ign switch.