Covering the hair is a completely different sugya; it’s not about tznius.
Of course it is – sa’ar b”isha ervah. The gezeiras hakosuv is needed to give it the issur, because it’s less of a taavah than other things, and is muttar for single girls, but at the end of the day, it’s still ervah, and is tznius related.
It’s also completely illogical to say that the full hair covering, which is das Yehudis and explicitly tznius based, is because of tznius, but the partial hair covering which is das Moshe, is not.