I suggested that you investigate rolling back because you said it was working prior to the Anniversary Update. I did some reading as well, and see that MS wants you to be running the Anniversary Update (release notes), but I wonder if its really needed. What about a wireless option. I thought the BT/Wireless controller was about $60. Probably more reliable too. System File Checker came back clean. That's good. On an aside, the BT device in DM... you can't roll it back to the previous driver?? Or, another option when uninstalling a device in DM.. you should be given the option (checkbox) to uninstall the driver for the device as well. So when you right-click and select uninstall from the context menu, you get the option to remove the driver as well. Reboot after and then try running the older installer. Failing that, you're at MS's mercy. At that point, I'd probably move to the cable, try wireless dongle or just return it. I'm like you. Give it a chance, take reasonable steps to make it work, but will return when you have tried and are having to spend hours jumping through hoops.
Low and behold after I posted this morning, one of my laptops got the Anniversary Update. The update blue screened while it was installing, then went into a crazy recovery. Took about 20 minutes, and I had to reactivate office. Seems to be fine now. I don't get my one S until next month. I haven't played Gears Of War for a long while. Kind of cool that you can play it on xbox and any W10 PC.