I reckon your skepticism was warranted, looked at the video some more and now have a better explanation i think. i updated the post.
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.
The Booster 9 static fire actually froze the ground
I'm pretty sure the white stuff that appeared on the ground is frost. Looking at it from this angle and other angles you can see the frost slowly forming and then melting again when the water hits it.
I think this is best explained by the intense shock waves decompressing the air, which causes it to cool down and take heat from the ground (of course they also compress the air by a lot, but with shock waves the time interval that the air is below atmospheric pressure can be much longer than the time interval that it is above atmospheric pressure). interestingly, there are also thermoacoustic refrigerators that function in a similar way.
Edit: I now think that cold gases from engine startup are more likely to have caused this, because as the static fire just started in the video you can see spots freezing over as those gas clouds approach them.
Taken from the overhead drone shot https://twitter.com/SpaceX/status/1695159316670718231