BYU is a unique place, no question. Does the HC help make it unique? Yes. But unfortunately I think it does more harm that good in that area. It's fine to be different, but how you're different matters. BYU has taken the wrong approach imo in being different in this way.
But take the HC away and BYU is still NOT just another school. If you can't see that, there's either something wrong with you, or you didn't go there.