How to uninstall MSDE on Windows 10?

I just upgraded my PC from Windows 7 to Windows 10. I did this because I need to upgrade SQL Server from 2014 to 2016 to match our production environment. I am unable to uninstall SQL Server 2014 due to a bug that seems to be due to SQL Server 2000 MSDE also being installed on my PC (I did not install this - can only assume it was bundled with some other software, maybe an old version of Visual Studio?)

The error message when attempting to uninstall SQL Server 2014 is:

SQL Server Setup has encountered the following error: '.', hexadecimal value 0x00, is an invalid character. Line 1, position 212550. Error code 0x84B10001.

The proposed fix for this issue is to uninstall MSDE. Unfortunately I cannot uninstall MSDE either. If I try to remove it via Control Panel, after clicking the UAC prompt, the dialog closes and nothing happens.

I found this page that claims to show how to remove MSDE manually if it won't install from Control Panel:

However none of this works either. The registry entries don't exist, so I cannot determine the product code from them, so I can't use MSIZAP.EXE. I managed to delete the files under Program Files as directed, and removed the registry entries for the services, but this does not fix the issue - Microsoft SQL Server Desktop Engine (SQL2000) is still list on Programs and Features on Control Panel, and I still can't uninstall SQL Server 2014.

I have attempted to install SSMS 17.3 only, but install fails with error code 0x80070643.

I think all I can do at this point is wipe my laptop and reinstall Windows 10 from scratch if I can't solve this.