As I recall, there are builds in the
NEWT catalog where twists are assembled around an inert core (insulating rope), presumably because proximity would be just too bad in the middle of that build, so why not keep current to the outside in the first place. They don't give any explanation of the various builds, but I assume these are for high frequencies relative to the diameter.
Indeed, skin effect doesn't stop just because of insulation; eventually the loss tangent of the insulation itself will dominate. Or on a related topic, ferrite cores are self-shielding in the same way (hysteresis makes up part of the large-signal loss tangent) and therefore up to a certain cross section / diameter is useful for a given frequency. (IIRC, typically some inches at 100kHz, so it's not a big deal, but still, part of the reason very large industrial transformers are made of bricks glued together.)
In modern context, the biggest thing is probably that, because Litz is transparent to magnetic fields (which is an equivalent description of what it does / how it works), the whole volume of the wire participates in leakage inductance. So it's rather ponderous to use at very low impedances, and foil/sheet may be preferable (despite the increased losses from eddy currents and such).
Conversely, because foil blocks fields, it's essentially useless for more than a single turn per section. So... application is a bit limited. Not to mention how hard it is to get connections in and out of a build.
Tim