Mmm… as long as the headphone plug and cable are not damaged, then the issue is within the logic board socket assembly. Apple uses its own design so you’ll need to find a junker with a good socket. Replacing the logic board is the very last option.
As to what went wrong there are two switches within the socket which the signals from the internal speaker/mic to the external ones. Here’s a deeper write up Understanding Audio Jack Switches and Schematics.
Sometimes you can use a toothpick to re-align the blades if twisted, I haven’t found this works in this series but worth a try
I think you might want to find a USB DAC unit which will by-pass the socket issue. Or just use a Bluetooth headphones/buds.
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