I found that the original on line update from 7 to 10 gave a flawed product. Mostly it took ages to boot and shut down (about 2 minutes), but it was not the fastest running OS in the world either.
My solution was, like yours, a new installation, but I formatted the C drive just before I did the install from a USB stick. And, of course, I had a full backup of all my files. This way, I figured, there would be no hangovers from the previous install either in the OS itself or the registry. It gave a clean system that booted and shutdown very quickly, about 7 - 10 seconds.
The downside of this method is that you will have to reinstall all your other software as well, then reinstall your data files from the backup.
At least you will have the opportunity to get rid of any bloatware and programs that looked like a "good idea" but you never actually use. Although W10 out of the box does tend to install a number of games you will probably want to delete.
If you don't know your product key there are several utilities that will show you it. For example, 6 are listed
here , I used Belarc Advisor.