So by now you are probably done with the Logitech support folks. As usual the manufactures don't hire anyone that has a clue.
The C920 worked fine on Windows 7 and early releases of Windows 10. Then Microsoft came along with the Creators update and presto no more reliable webcam. It sometimes works, for a minute or so and then disappears. Logitech support will have you do the usual handstands and it won't solve the problem.
The fix is simple enough provided you have a USB 2 port. Don't plug your C920 into a USB 3 port. It won't work reliably with USB 3 and Windows 10. Once I moved my C920 to a USB 2 port it's back to normal.