Why does a university not require me to send proof of high school degree?
This is not normal in the US -- but that doesn't mean it's an illegitimate process. Private institutions can select applications using whatever criteria they deem appropriate; so while it sounds a bit questionable to me, it's perfectly within reason.

It is also perfectly valid to admit students without a high school degree; it happens to be uncommon, but depending on the subject matter of the BA, it may not be (at least in the university's eyes) a strict necessity, provided your other credentials are sound.