For better or worse, the web has always been driven by specifications, rather than implementations. Properly standardizing all of SQLite in a spec is not trivial, and as much I think SQLite is great software, I don't think that relying on SQLite the library specifically would be a good idea. What if the latest version of SQLite does something that's not in the spec or prohibited by the spec?
Either way, there's IndexedDB now, which is supported by almost everything, including IE.