My app does end-to-end encryption, and this is much more straightforward in a "real" app. (1) I need non-volatile storage for the keys, and (2) I want users to know if/when they get a new version of the code.

You can do E2EE in a web app (like Element.io), but then every time you load the page, you're trusting the server not to send you a new version with a backdoor.