Yes, "full" internet access is a hard to define term and this simple test doesn't cover all possible cases. But it will work in probably 99.99% of cases and honestly I think that's good enough. If it doesn't work you end up with a little "no internet connection" icon in your taskbar and that's about it. You can still use the connection, so I don't think it's a big deal that this test isn't 100% accurate. It's still more useful to the majority of users to have a slightly inaccurate test than to have no test at all.