OMG. Thank you thank you thank you. That is so simple, and phenomenally careless on Apple's part. What is their problem???
For the record, I will mention that this problem happens not only on El Capitan, but also on Yosemite with Safari 9. At the top of the "bad" Extensions Gallery page, I see this lovely message:
Extensions in this gallery are compatible with Safari 9.0 or later on OS X El Capitan or Yosemite. Please upgrade to
OS X El Capitan which includes Safari 9.0, or update to OS X Yosemite 10.10.5 and install Safari 9.0 from the Mac App Store.
Well, I'm running 10.10.5 and Safari 9.0.3. Clearly, something is wrong in either Safari 9 or the safari-extensions.apple.com page that is preventing the page from correctly identifying the OS and browser. (This message also pops up if you visit the Extensions Gallery in a browser other than Safari 9 or on an OS other than Yosemite or El Capitan.) It's clearly not simply a problem with El Capitan's new security features.
Hopefully this extra info can help bring about a more permanent fix from Apple.