Increasing our use of the Air Force as a provider of humanitarian aid would allow us to preempt a terrorist foothold and create a positive view of the US (similar to our use of positive propaganda in the Cold War).
We'd need the AF to do a lot more work than just treat boo-boos and fill a few cavities. It's not that I disagree with your point, just that we are involved in strategic engagement all the time in Africa. Just look at the example of JTF-HOA.

Sometimes these initiatives can plant seeds that grow well, but these engagement efforts are by no means a guarantee for success. Could the AF be more involved? I don't know enough about its operational and exercise footprint to say yes or no. I do believe that the sense of purpose needs to be ingrained into it culturally, in order for it to remain relevant in times of Small Wars though. You don't get much of that, however, when you are worrying about the infrastructure, training, personnel, and O&M needed for a new fleet of Raptors.