So, the "just let it burn" strategy when fires start might
  lead to a lot more property damage than a strategy of
  proactive burning.

The U.S. is _huge_. There's too much land to do controlled burning. And an intentionally set fire is morally perilous, so there are strong incentives to avoid burning except where it's easy and obvious; that is, _away_ from housing or in only very small patches to create fire breaks.

  Also, from a carbon footprint point of view, it seems
  advantageous to mix burning and logging, so that not all the
  carbon returns straight into the atmosphere.

But loggers are gonna take the good trees and leave behind the young ones, precisely the opposite of what a fire would do. I know this isn't how it's supposed to work on paper, but the incentives aren't well aligned. A natural fire isn't going to lobby Congress to increase its quota of high-value trees.

I think any practical land management plan that has a shot at restoring forest health requires letting many more natural fires burn, and to burn more widely. The dilemma is that things are so bad that these fires are much more dangerous than they'd normally be. But I don't see how you ever get over that hump unless you can send out an armada of autonomous, self-replicating robots to cut down the underbrush. Perhaps we'll just have to wait for that solution.