Ah, my misreading of that post.
The device that causes the flashing of the turn and hazard signal lights, when either are being used, is the Turn Signal and Hazard Module. It's mounted on the side of the steering column and shown as #10 in
http://opposedforces.com/parts/lega...ctronic/electrical_parts_body/illustration_1/
If the module is removed, that would disconnect the hazard switch from its downstream turn signal system wiring. As there is something in the turn signal or hazard signals system that seems to be related to the engine starting, perhaps remove the module and see if that resolves the problem.
If it doesn't, then remove the connector from the back of the Hazard switch. With the switch isolated, it's position would be irrelevant. See if this allows starting.
The question that this tries to address is why the starter doesn't work when the hazard switch is off, but is fine when the hazard lights are turned on. There is a roundabout link between the starter circuit and the hazard switch and turn signals system. As you have the wiring diagrams, four sections are relevant: Power Supply Routing, Turn Signal Light and Hazard Light System, Security System, and Starter System. Briefly, if there's an unusual load placed on the power coming to the hazard switch from the ignition relay, it could also cause the security system to not activate the security interrupt relay in the starter system, and that would prevent starting. Granted, this is all somewhat stretching it . . .