Just so you know, as an inductor is just a small coil of copper, the resistivity might be low enough to trigger the beeping on the multimeter! So maybe it’s not dead. I’ve seen a lot of burned down inductor and trust me it looks way worse.

If your multimeter can measure resistance in miliHoms, you should be able to measure at least ~10mR or more (that order of magnitude) Note that measuring a circuit like so is not guaranteed since there’s other stuff on a PCB that gets measured at the same time.

Did you remove the motherboard and looked at the other side?