Since I had the assumption yesterday at Stop Market Yagur, that the reason the contorl gear of induction lamps, called "Generator" (HF generator), instead of "Driver (Like the case of LED)" or "Ballast (Like the case of discharge lamps)", is because unlike electronic fluorescent ballast which regulate current and voltage to the lamp in addition to convert the mains frequency to 20,000hz, HF generators of induction lamps, simply generates high frequency AC to the lamp induction coils and nothing else (As no electrical current or voltage passes through the lamp itself that needs to be regulated), a thinking came to my mind that induction lamps could theoretically operates directly from the mains without an additional gear if no HF would be needed for a good coupling efficiency and 50-60hz would be sufficient (Much like incandescent and halogen lamps).
Your thinking?

With any lamp you don't need any ballast, when the source characteristics matches what the lamp needs. Otherwise everything between could be called "Auxiliary", "Ballast", "Generator", "Driver" or so.

For the induction and mains:
A bit of nitpicking: As the induction lamp normally operates on AC magnetic field, it won't operate form the mains, as that delivers just the electric voltage/current. So you need at least the coupler, what in fact is a form of "Auxiliary", "Ballast", "Generator", "Driver" or whatever you want to call that.

Back to the question: The induction need sufficient electrical field to be build by the induction along the discharge path for the arc to burn. So the field inside the single turn secondary (= the arc) has to be so strong, the induced voltage is sufficient to form such field along the discharge path length.
And the induction law tells, the voltage formed in one turn is a d(flux)/d(time). That means just fast changing magnetic flux, nothing else.
And if the flux has to change so fast, it should either be allowed to reverse the direction very frequently, so it does not have to grow to high values, or it should be allowed to grow to just large values.
That would require either a strong field, comparable with those found in nuclear explosion (well, not as much practical), or you would need very high cross section of the magnetic core. But the high magnetic core cross section means the arc would become longer as well, so need higher voltage. So operating at 50Hz instead of 250kHz with the same flux density and the same dimensional proportions mean the cross section would have to be (5000^2) as large, arc length 5000x larger and with the same power density (although that would have to increase with larger cross sections to retain the efficacy, here I'm neglecting that) it means (5000^3) times higher power rating of such lamp. So instead of let say a 30W lamp (the minimum power of the Icetron/Endura concept I've seen) you would end up with a lamp of about 3750GW power input...
Well, I won't call that really practical...