Does the car start in Park but not Neutral? If so the switch is probably OK but out of adjustment. If it does not start in either P or N you could temporarily use a jumper wire to connect the start interlock terminals on the switch together and see if it starts that way. If it does the switch is probably bad.
Since the 57 was built off the 56 Stude President model using that engine and trans I am not sure if it follows the illustrations for starter circuits on 56 models mentioned in the Stude service manual. That manual (fig 32, page 20, electrical section) http://www.packardinfo.com/xoops/html ... nual1956/Electrical_7.pdf
shows a totally different interlock circuit than what I am used to seeing on Packards. Not a hundred percent sure the drawing is correct though based on the description of the circuit a paragraph or so before the illustration.
If the drawing is correct it would seem they interrupt battery power going to the ign switch rather than Packards interruption of only the signal from the starter switch to the solenoid. Not quite sure how the circuit works to have the ign switch powered when in gear unless it gets power from other wires connected to the A terminal of the switch but if the safety switch does carry a significant amount of current then possibly the switch was not making completely at the Neutral position and the current going thru a relatively small contact area burned the switch. If that is the case then replacement would be the best option.
I believe the steering column may be similar on the 57 but can't say if the switch is in the same position or the same as used on 55-56 Packard models. On the Packard models the switch is also used for the reverse light and there is a delicate point where you can get all three positions to activate the switch properly. In either situation wear in the connection to the shift tube or if the screws holding the switch to the column loosened slightly and allowed the switch to move could be an issue. If the drawing is wrong and the switch is actually in the wire between the start command and solenoid as Packard did it then I would first try adjustment or see if there is excessive play in the actuator to shift tube connection.
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