Electronic – Inductor with Ground Shield

My guess is that the inductor is probably not a major contributing factor to the failures. I say this because it's a inductor, and as such produces a magnetic field. The quartz in the crystal is not particularly magnetic. Second, you note that the inductor is shielded, so the external leakage field should be small. Still, the crystal output is pretty high impedance and induction could be adding a small voltage in series with it. This should be small and not much issue if the crystal is being driven properly with the right load caps.

I would look closely at the crystal circuit. This sounds like maybe you are using too low of a drive level setting or the load caps aren't right. What does the datasheet say the load capacitance of the crystal needs to be? What size cap do you have on each side? The chip driving the crystal may have different drive level settings if it's intended to run with a wide range of crystals. At 24 MHz you almost certainly should be using the highest drive level choice.

Look at the crystal driver output pin (input to crystal) with a high impedance probe. That should be a good strong signal. It could be sortof square wave looking with a bit rounded corners or nearly a sine wave. It should be a few Volts p-p, usually half the supply voltage or more. Check the frequency carefully to make sure it's not running on a harmonic. If it is, it's a indication one or both load caps aren't connected or are too low. Then look at the crystal output. That should be a nice 24 MHz sine wave at least a Volt p-p, preferably a little more. Does the unit suddenly work right when you put the scope probe on a crystal lead? If so, it's again missing, bad, or incorrect load caps.

You should always use a current limiter (typically about 50 mA) to feed a bias tee for a GPS antenna. There are just too many ways to create a temporary short when connecting cables, etc.

It sounds like your antenna has a faulty cable or some other internal fault that causes intermittent shorts.