I created a Firefox extension that can check signatures both in the certificate and (in the meantime) hardcoded in the extension. [2]
[1]: http://blog.airbornos.com/post/2015/04/25/Secure-Delivery-of...
[2]: http://blog.airbornos.com/post/2016/02/23/The-first-web-appl...
And it does not necessarily address the compromise of the hoster either, since the attacker who has control over that may also have access to the signing keys. Especially in the case of a court order.
That effectively is a silent, user-targetable auto-update of software to which you entrust your secret data.
Auto-updates for crypto software should be verifiable through other channels, require user confirmation and use a distribution method that is difficult to scope to a single user to make targeted attacks harder.
It's true that these are not great against a state level actor deliberately targetting an individual (but then, not much is), but e.g. in a lot of situations I would trust github to serve me unmodified html and s3 to store my encrypted data, where I might have not trusted a single entity to provide me with an integrated solution.
It would be great though if a page could indicate that it is sensitive and then if the browser sees it change it pops up a message on the client giving the user a fingerprint and notifying them of the change.
SRI means you don't have to worry about your js getting modified so all that's left is the html. My claim is that there are lots of situations where I can trust something to serve html honestly enough for my use.
It is possible that my bias for web apps that connect to real services (of the sort a desktop app might) rather than as a thin veneer over a black box service gives us a different perspective.
But I do ultimately agree that some support for enhancing trust in the initial html document should be built into browsers. I guess such a thing should involve being able to get a hash of the actually downloaded and running code and document and being able to postpone an update until I've had the opportunity to check it's legitimate.
The page itself can still be modified, but there are mitigations that can be performed if you're paranoid enough.
1. https://developer.mozilla.org/en-US/docs/Web/Security/Subres...