“I think R’ Moshe is learning (and the lashon of the Rambam sounds this way) that the added kedushah is not b’etzem, it’s a product of the extra chiyuvim. Kohanim, OTOH, have their extra mitzvos because of the higher level of kedushah.”
I don’t think that’s correct. R’ Moshe says explicity the opposite of that. See here:
“The mitzvah of You shall be holy, which is followed by a recitation of several of the fundamental mitzvos, is not of the same type as the mitzvos that follow it. This mitzvah means that every Jew should realize that he is sanctified with the holiness of the Jew, and it is only because of that holiness that we were given the Torah and obligated to do the mitzvos. As I have often written, mitzvos cannot be fulfilled properly unless the doer has the holiness of the Jew. The Kohanim, who have additional mitzvos, must have the particular holiness of Kohanim. This is why we make a blessing before mitzvos and say, “Who has sanctified us with His mitzvos”; and Kohanim, before doing mitzvos that are limited to Kohanim, say, “Who has sanctified us with the sanctity of Aharon.” The expression “Who has sanctified us with His mitzvos” should not be misunderstood as meaning that mitzvos are the source of the sanctity. It is self-understood that the sanctity the blessing refers to is the underlying sanctity of every Jew — that which enables us to fulfill the mitzvos.”
Rav Moshe Feinstein, zt’l, Darash Moshe, Volume II, p. 154, Vayikra, Kedoshim