I don't know if this applies to Windows 7, but it does work in Windows 8 and Windows 10.
- Open "RegEdit"
- Go to "HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows NT\CurrentVersion\NetworkList\Profiles"
- Locate the Network Location
- Change the DWORD Value "Category" to 1
If this doesn't work, delete the key of the network, restart your computer, and repeat. It should work after that.
Not exactly the answer, but this led to the solution. FINALLY FIXED!! Here's how:
In RegEdit, at "HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows NT\CurrentVersion\NetworkList\Profiles" I had three profiles with names such as {4A3BFF36-4EA6-4415-AA5F-46CE449C95AB}. I tried several things including changing the DWORD value of the item "Category" to 1 in all three of these profiles. That did NOT work.
However, I eventually deleted all three of the Profiles.
Also, at "HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows NT\CurrentVersion\NetworkList\Signatures\Unmanaged" were three corresponding items which I also deleted. This cleaned the slate, so to speak.
THIS WORKED!
When I re-booted, I was able to change my network from Public to Private and it walked me through the process of joining an existing Home Group established by another computer in our household.
Finally nearly 3 years after my original post, I have the problem fixed.
Many thanks to BennyPootron for the insight to a solution.