Ok, so I have done all the testing I can, and come up with some interesting results.
I started by trying different browsers. I am on Windows 8.1, and I tried Google Chrome, Firefox, and IE, each at their latest stable version. The site did not update on any of them.
I then tried on Linux, specifically Arch, using Chromium, and Firefox. The site did not work on either.
I then switched back to windows, and tried switching dns servers, still no luck (not that I expected anything).
I tried on my laptop, both on windows 7 and linux, no luck on either.
I even tried on my Android tablet, no luck on there.
Then, I tried a VPN to the US. Finally, FINALLY! The site works as intended. Things change on the page without having to refresh! Unfortunately, any server I connect to would have at least 200 ping (live in NZ), so the game would be unplayable, but at least we are getting somewhere.
So, the question is: Why does the VPN fix things? Is my ISP having some shenanigans with my connection to TF2Center? How can I troubleshoot this further?