My solution to re-installing a fresh Windows 10 would be to download the ‘Windows 10 Media Creation Tool’ from Microsoft and create a Windows 10 bootable USB stick. I would take the opportunity to replace the original hard drive with a fast SSD drive and keep your original one to one side so as not to lose anything.
SSD Drives can be bought for £28 now.
When you install windows fresh from the USB stick you don’t need a product number to complete the installation as it should recognise you hardware and activate the new install.
When you make the Windows bootable media disk (USB) you select create media
Download the media creation tool from the microsoft link below
https://www.microsoft.com/en-gb/software-download/windows10
Follow the instruction when running the creation tool.
You mention that your motherboard is only 10 years old and though you should get some future proof from it, well in all honesty tech moves on very quickly, while there is no reason you cannot use an old board as long as it meets (WIN10 requirements) but some parts in your system do not meet requirements. In the industry of PC parts 1 year old can be considered out of date at times.