Did Woodward buy his "religion reporter" credentials at a fire sale? If he thinks Mormons haven't been willing to share their faith, he's not paying attention.
I'm not Morman, so I speak from significant ignorance. But, from my understanding with friends and co-workers who are Mormom, those "many veils" involve practices of the church for members. Become a member, you'll find out what's there. Whether I agree with the LDS church or not, that makes sense to me.
Totally off-topic: One of my favorite movie lines,
Jim Kirk to Jillian in Star Trek IV: The Voyage Home, referring to Spock, Kirk says "He was at Berkeley in the 60s, he did a little too much LDS."
You may be interested in the expose' book of former Mormon Deborah Lake's Secret Ceremony(1992-3??). Then again you may not find it as the 'missionaries' have been given permission to steal the book from bookstores. From what I can discern from former Mormons...this author's experiences with the Mormon hierarchy, practices parallels their own.