Regarding White Mountain, I camped at Grandview last year before hiking it. It was pretty nice, good tent sites, moderate amount of trees, not many people, beautiful views. As I recall, it was about an hour drive from there to Barcroft Gate.
When we arrived at Barcroft Gate, I remember thinking "I'm really glad we didn't try to camp here." It's basically just an open plain, with no trees or anything, and nothing to block the high winds. It looked like a miserable place to camp.
