The Web folder is for the Chromium Embedded Framework (CEF) which is for the Chromium engine that Media Center can use after it downloads the components. And you're right, Media Center also relies on WebView2 for the Microsoft Edge engine which is the default web browser engine in Media Center because it's the only one that works with DRM and whatnot, at least it did at the time when Hendrik added support for it, if I recall correctly. The Internet Explorer engine option is still there, but as of yesterday Microsoft has fully killed all support for Internet Explorer so it *may* be time to remove that option, even though it likely still works. It's probably a security risk keeping it around and using it, IMO.
Now the question is, why is WebView2 not working for you and why does it only work if you start Media Center as an admin? Perhaps it's some sort of permissions issue? What version of Windows are you using? Are you up-to-date on Windows updates? Do you have a second PC or anything to test it on? If I had to guess, something's broken with your WebView2 on your machine, because I haven't seen anyone else report anything like this here... yet.
Anyone else using WebView2 via the Microsoft Edge engine in Media Center having similar issues?
P.S. As far as I know it should be included with Windows (as I'm not sure if MC will download it if missing) but you can manually download and install WebView2 from here:
https://developer.microsoft.com/en-us/microsoft-edge/webview2/ and Windows should keep it updated in the background via Windows Update.