I want to reinstall Windows 10 from scratch on one of these
workstation, but I don't want to have to install windows 7 (or 8)
Just install Windows 10. Since the machine in question already has an activated copy of Windows 10, it will automatically activate after you reinstall. When asked for the product key you can either skip that step or provide the original license key from Windows 7 or Windows 8 that you used.
Is there an inherent Microsoft-Exclusive way of obtaining Windows 10
Activation keys from existing installations?
If you upgraded from a previous installation of Windows there isn't an activation key that can be extracted. Here is Microsoft's official stance on the subject though:
Activation in Windows 10
Activation helps verify that your copy of Windows is genuine and
hasn’t been used on more devices than the Microsoft Software License
Terms allow. Depending on how you got your copy of Windows 10,
activation will use either a digital entitlement or a 25-character
Digital entitlement is a new method of activation in Windows 10 that
doesn't require you to enter a product key.
The article continues with the following notice:
Starting with the November update, Windows 10 (Version 1511) can be
activated using some Windows 7, Windows 8, and Windows 8.1 product
keys. For more info, see the section Activating Windows 10 (Version
1511 or higher) using a Windows 7, Windows 8, or Windows 8.1 product
key in this topic.
The following applies to the question's described situation:
How you got Windows 10:
You upgraded to Windows 10 for free from an eligible device running a genuine copy of Windows 7 or Windows 8.1.
You upgraded to Windows 10 (Version 1511 or higher) and activated it using a Windows 7, Windows 8, or Windows 8.1 product key.
- Digital license (called a digital entitlement in Windows 10 Version 1511) is a method of activation in Windows 10 that doesn't require you to enter a product key.
- (A digital entitlement will be given to your device for Windows 10 based on the valid product key you entered)
Windows 10 Activation
What this basically means is that any machine that has ever had an activated Windows 10 installation on it, will automatically activated upon future installations of Windows 10, even if a product key is not provided upon the installation of Windows 10. If installing onto a formatted drive there is of course no way to get the "upgrade" digital entitlement, but a valid key from an older Windows license will work instead.
It is worth making it clear that, Windows users who accept the free Windows 10 license, are not granted their own unique Windows 10 product key.