It was standard practice to run the 'neutral' or 'identified' conductor un-switched to all the components.  Often, the manufacturers just ran one wire from the connection to the power cord all the way through. Nothing wrong with that.

It is correct that the 'hot' or 'line' side should be run through switches, but I've also seen 1950's and earlier appliances in which this was not the case. Your drawing would work, though UL wouldn't have liked it even then.

It's unfortunate your original wiring needs replacement. Most manufacturers tended to accept a slightly longer run of wire than strictly needed in order not to cause interference with the insulation value or to create paths for condensation to soak the insulation, thus ruining it.

If the wiring underneath is damaged, the wiring inside the case may be, too. Or, it may be in perfect condition - you'll not know until you look. I've never been able to 'fish' new wiring through by hooking it onto the old and pulling, but many here have done so successfully.

I suggest either a grounded line cord or a GFCI outlet for this refrigerator. Congratulations, on a neat find. More pics, please!

so worth repairing. Thanks for sharing. Keep sending pics as you progress or post any questions.

Thanks!

Looks good!