I agree, voltage the same on both sides of the connector means it isn't that connector.
it could be the ignition switch, as you noted due to the voltage drop there.
I would see if the switch has any resistance between the terminals disconnected from power.
you also might pull the neutral safety from the starter relay and test the voltage in start and see if that is down too.
I would be tempted to jumper those terminals on the switch and see what the voltage does while the switch is on as it was in his previous tests. If it magically jumps up the switch is likely the culprit. DO VERIFY the terminals at the connector are not the issue.
I'm not sure the resistance check will tell him much as there is no "LOAD" on the switch, so it may indicate little to no resistance but may not be able to pass the current needed for the load thereby causing the V drop.
keep us posted
