Yes, model it in a simulator. That question is designed to highlight the strengths and limitations of mathemetical modelling, simulations, and solderless breadboards.
I will be honest I personally would not know how to model this particular scenario in a simulator. However having read some of the previous posts I would probably do it with a capacitor in series with the bulb. Would that be the correct way?
The key point is that the circuit works (or fails!) because of "unexpected and invisible" parasitic components. Minimising those is part of sound circuit design; almost always the intention is that the circuit's operation is determined by well-defined components. No surprise there!
There are, in general, many parasitic "components" most of which are probably unimportant. In this circuit the important parasitic component that allows the effect to be visible is a capacitance in series with the neon and voltage source. Hence the effect could be demonstrated by a model containing the AC voltage source, capacitor, neon, plus resistor representing the internal source resistance of the voltage source.
It would be non-trivial to choose the capacitance value, and given the very strange piece of test equipment I used as a voltage source, determining appropriate values for the voltage amplitude and resistance wouldn't be trivial.
I wrongly assumed that it was inductance that was a problem but now that you elaborated what is happening it makes sense. I knew the metal clips would be involved somehow and it makes sense that there would be capacitance based on the air gap which in this case would be the dielectric.
Having a good mental model of what's present and what's happening leads to such understanding. Having a knowledge of theoretical models and behaviour is a pre-requisite for making sense when an implementation doesn't behave as expected. Simulation can sometimes be used to confirm the cause of observed behaviour.
As embarrassing as it is I am not familiar with the neon bulb. I do not know the current requirements to light one versus incandescent that is why I stated incandescent.
My follow-up question is will this work with an LED since its current requirements are very small? My guess is it would.
Don't be embarassed; neon bulbs are largely obsolete. The nearest component used in modern equipment is a gas discharge tube used to protect against high voltage transients on PSU inputs.
They were used for strange things; before I sold it I had a Fluke DVM that used a neon as the reference voltage. Since it wasn't very stable, it had a front-panel control that enabled the displayed value to be tweaked until it showed 1.019V, the same as a (stable) Weston standard cell. You can also implement digital logic with them, but I wouldn't recommend it!
Neons require 60-120V to illuminate, but the current is 10s of microamps. Thus a LED might illuminate, but not brightly.
I also assume you used some sort of switch mode power supply due to the 150 kHz mentioned.
The voltage source was just something I had to hand, a strange device used for testing bulbs, a TIS1040. It is more an oscillator than an SMPS.