There are plenty of active commentators in the Mormon blogosphere—termed the bloggernacle—who opine about this sort of thing daily. Or lots of Mormons who would no doubt be happy to be asked—Baker, Williams, and Givens all were.
There have been a few good examples of how this can be done in a way that both informs readers and humanizes Mormons without advocating for them. See this story from The New York Times, this one from Politico, and the many written by McKay Coppins (who also happens to be a Mormon) at Buzzfeed. Weird it hasn’t been tried more before.
So, what do Mormons think of ‘their’ moment? They think the media, and particularly the commentator class, is missing it.