Reb Moshe says we have mitzvos because we have holiness, not the reverse. But he says men and women are equal in holiness.
“You should know that all of this (the exemption of women from positive time bound commandments) is not because women are on a lower level of holiness than are men. While obligation in commandments results strictly from one’s having holiness, men and women, in that sense, are equal in holiness. All the verses in the Torah regarding holiness refer also to women. (This applies from) the beginning (with) the arrangement to receive the Torah (at Sinai). “You’ll be to me a treasure and you’ll be to me a holy nation.” Exodus 19:6. (The subject of this verse is all of the nation of Israel) as it says “house of Jacob” in referring to the women and “tell to the house of Israel” in referring to the men. Exodus 19:3. “
Igros Moshe, Orach Chaim IV #49