“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.”
With the Rambam it’s hard to tell. With Reb Moshe it seems pretty clear. He says the shaychas of the chiyuv of mitzvos is FROM kiddushah. He doesn’t say kiddushah is from mitzvos.
He might get this from the Gemara in Yavamos that considers that an extra pasuk is need to learn a mitzvah for cohanim because they have more kiddushah.
However, he notes men and women are equal in kiddushah. This is why we say women are exempt. They have a connection to the mitzvah, but are exempt, (like a chason on his wedding day is exempt from krias shema-my note). This is why they say a bracha when doing lulav et al.