I tried the iOS 9.0.1 update. I thought it fixed the problem. (Yes, the exact same problem, but mine seems worse when I'm coming back to an app using the new app switcher, but it happens on original launch and when I try to get back to an open app from the home screen. And, no, it doesn't matter how long you wait the screen doesn't "unblack.") It seemed to go away for a day, but it's back with a vengeance this morning. Restarts do seem to make the problem disappear for a while, but it comes back.
I, too, have tried resetting and reloading. It does not help. I normally do not allow my phone to provide diagnostics info, but I turned it on, now. Looking through the log files in diagnostics, I don't see anything to indicate that these errors are caught and logged. Hopefully, Apple is reading these posts and attempting to solve this. We have four iPhones in the family. Two have this problem (both are iPhone 5 devices, 64 GB and 16GB). They happen to be the only ones that have been updated with iOS 9. Coincidence? I think not.
Best of luck solving this, but this isn't our fault, and I suspect it's out of our control to come up with a fix or work-around. This is iOS based, and most likely beyond us in user land.
Here's hoping for iOS 9.0.2 ...