yes, although adiabatic just means that there is no heat transfer between the gas and its environment, so the gas itself could still change temperature in an adiabatic process. but yeah if there's no work done by the gas then it doesn't change temperature.
at this point i'm quite convinced but i'd love to hear an alternate explanation for the frost
as threelonmuseteers said, as long as the gas does some kind of work while expanding (which could just be exerting a force on the surrounding air), it will expend energy which will cause it to cool down.
I reckon your skepticism was warranted, looked at the video some more and now have a better explanation i think. i updated the post.