I'm guessing certain icons are in the Windows icon cache because of some other program you used to create shortcuts (IE for example, but good be others I guess) and Windows keeps a URL reference for each icon and applies them to shortcuts that don't specify their own icons. Try with a site you've never visited before in any browser or any app and create a shortcut for it. You probably won't get the site's icon for it.
If you use some utility to clear the Windows icon cache, I'm guessing that none of the shortcuts on that system that you produce with Opera will show the site's icon.
I guess I could do that each time I want to save a link to a website but it's incredibly clunky!
Yeah, sounds like that'd work. But, as you said, that'd be clunky.
I wonder why they don't want to fix this. I mean, it's not like web browsing is a new thing.
I don't know, but since Opera is Chromium-based, if you get this improved/fixed in Chromium, all Chromium-based browsers including Opera should get it too.