It only breaks the web if you cut everything over from http to https. If you can serve both you don't have a problem.
This works fine if you use anchors without protocols in your html:
a href="//site.com/resource"
This works fine if you use anchors without protocols in your html:
a href="//site.com/resource"
It's a simple fix to whitelist the one website though. It doesn't break the Internet. It breaks sometimes, for some users, and is trivially fixable when it does break.
"Fundamentally breaking the internet", to me, is something that actually breaks the usability of the internet in a non-trivial-to-fix way for the end user where the end user isn't even in control of the fix. That's breaking the web.
Failing to support IE5 is "breaking the web" in the same way Https Everywhere breaks the web. In a way that is to be fixed on the user-end.
(Although sites that fail to serve over https:// should fix their site)
HTTP and HTTPS use different ports (TCP 80 and 443). You can run one web server application on the HTTP port serving content A, and a completely different web server application on the HTTPS port serving content B. With firewalls doing NAT port translation this could even result in HTTP requests going to a completely different machine than the HTTPS requests.
From a non-tech reason, there are some types of sites that the HTTPS content should never be available on a HTTP site. For example an online payment form. In such cases a sensible website will either disable HTTP entirely (and use a subdomain for secure content, rather than the top leve/ www. domain), or have a basic HTTP site that transparently redirects to the HTTPS version.
It only takes 1 server to serve differing content through each protocol to break things for the user
(I do agree that it's contrary to expectations, but I'm sure this doesn't stop sites from relying on it.)