The latter depends very strongly on how much computation is needed to compute those embedding vectors.

If you run a GPT-3.5-sized mode to compute that embedding (which would be a bit absurd, but if you really want GPT-3.5-quality classification, you may well be doing something like this), you're looking through quite a few tens of billions of parameters and doing a correspondingly large number of FLOPs, which could be just as expensive as running gzip over your whole (small, private) training set.