Power supply filter choke failure - how and why?
Quite likely, the choke isn't broken. It is normal for chokes to act as speakers, especially under high momentary current (think of transformer hum on loaded transformers). So your choke got noisy, because there are irregular current peaks across it. You observed the irregularities on your scope.

The most likely reason for erratic voltage at the choke (causing erratic current) is instability of the regulator. The idea of a switch-mode buck regulator is to draw current spikes at a high voltage and provide continuous current at low voltage. In your design, the current spikes should be delivered from C1/C2, and the capacitors should get smoothly recharged through the choke. If C1/C2 fail to source the required current, the input voltage of the regulator breaks down, up to the point where the regulator stops working. The external supply can not source the required current spike, because of the coil impedance at high frequencies.

If you replace the coil by a shunt, you lower the impedance if the external power supply, so the output filter cap can substitute C1/C2, at the cost of high-frequency components on the supply cable, so the modified device is likely to violate EMI regulations.

The proper fix in that case is to replace the input caps by new, proper low-ESR (important!) caps. To check the health of those caps, a capacitance measurement is insufficient; you need an impedance or ESR test.