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.
What? The specifications document what browsers already do. Have you seen the HTML5 spec? It's mostly formalization of error-recovery, showing how to interpret previously technically-broken documents in the way modern browsers have converged on. It's not what anyone would design, but there's no one driving.
Surely you're joking ? Having things done by specifications first has been extremly rare, pretty much everything rather followed the "make the specification after the fact to be whatever X browser has done and is used". Protocols, markup, css, javascript, the whole thing was done by implementation.
The people who rejected WebSQL proposal had a lack of decisiveness and technically-focused vision. They also interpreted everything very politically.