Hi! I believe I may have a solution to our problem of the black screen with the spinning circle.
I've stumbled upon this post today because my brand new iPhone 7 (which I have upgraded to the latest ios) also got stuck on a black screen with the spinning circle. I was trying to update some of my apps, when the screen suddenly froze. I clicked the home button once, and the screen became black with the subsequent appearance of the spinning circle.
I tried everything - pressed simultaneously the home and power button (did not work); connected it to itunes to do a restore (it won't let me because my "Find my iPhone" was turned on - but I couldn't turn it off because my phone was unresponsive).
Anyway, a very helpful Mr. Damon (a senior advisor from Apple) helped me via chat to resolve the problem.
Here's what I did:
1. Make sure your iPhone is unplugged from the socket or the computer.
2. Hold down the POWER button plus the VOLUME DOWN button SIMULTANEOUSLY. If it works, the screen should power off almost immediately.
3. Connect your iPhone to the computer. (when I connected to the computer, my iPhone automatically turned on and was ready to be used again).
The subsequent steps below are for those whose iPhones did not turn on with the above steps. The following steps are designed for you to be able to remotely turn off your "Find my iPhone", so you can do a manual restore from iTunes.
4. Go to www.icloud.com.
5. Click Find iPhone
6. Since the phone is powered off already, it won't be able to locate it. Click DEVICES on the upper bar.
7. Click your iPhone.
8. Click "remove from account".
9. Go to your itunes, and do a restore.
I did not have to do steps 4-9 anymore, because as I have mentioned, my iPhone automatically turned on after step 3.
This really worked for me. Hope it works for everyone else too! Have a great day!