[rickst29]

Here's what you had:

If the "black and "green" segments are undisturbed and were unable to receive electric current from white wire), then voltage difference between the wires could show full voltage, even with just one strand intact along the white wire (through the break).

But with most of the white wire strands copper damaged and unable to carry larger current (through damaged segment), the remaining portion was unable to support all the current which the the AC unit (or its heater strip) needed to receive when powered up.

The tiny portion of wire which was functional became overheated (creating a fire hazard), and the "voltage drop" through a few overloaded strands was to much for the AC to start up or run.
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There are, however, issues which concern me with your photo of the broken cable. The outside insulation appears super thin, and the 3 insulated wires were NOT assembled into the cable with additional fibrous materials stranded into the cable(along with the twisting wires) in order to fill" the round center of the cable evenly.

That extra "fill" prevents movement and scraping of the main conductor wires against each other, and against the outer insulation jacket. The outside insulation is also too thin, that's a mere "300 volt" cable assembly which should not be used with sun, weather, wind damaging the insulation.

The label on the type of cable you should be using is "SEOOW". That's a 600 volt cable with superior outer insulation (sun resistant and weatherproof), and it includes the required fill. Compare your cable to this photo of an the partly stripped end of an SEOOW cable, and you'll see the differences.

Unfortunately, SEOOW cable is bit hard to find in short lengths, and it's costly.