There's a much larger than zero possibility something (network gear) North of this thing is black-holing ICMP echo and/or your ISP is dropping the responses.

The site's IP is announced by https://www.blix.com/. It's conceivable that an IBM 5170 is sitting on its side in a colo rack, with a transceiver bridging its Ethernet card directly to the ISP's network connection.

But it seems more likely traffic is being forwarded or even rewritten at Layer 4. I don't think running something like HAProxy in non-caching mode would invalidate the website's claims. Even an IP forwarding solution (eg a GRE tunnel) might have http://shouldiblockicmp.com/ issues.