WebSQL as implemented was a terrible idea. Simply exposing SQLite's dialect of SQL to the web was always a non-starter in terms of web standards. You will quickly find (as you always do on the web) that pages would depend on bugs and other quirks in a particular version of SQLite, so trying to update the version you ship would be painful. You could ship multiple versions, or some sort of IE-like compatibility hell, or you could expose the version number and force web developers to test for ever possible version of SQLite so they can handle quirks themselves (protip: they won't).
The only way it was ever going to get off the ground was if someone fully spec'ed out a dialect of SQL that browsers could implement without being tied to the specifics of how SQLite works. Nobody stepped up to do that (and browser vendors weren't exactly foaming at the mouth to write their own SQL engine), so it died. You'd be better off writing a JS SQL engine that worked on top of IndexedDB these days, given that that's shipping in all major browsers.
The only way it was ever going to get off the ground was if someone fully spec'ed out a dialect of SQL that browsers could implement without being tied to the specifics of how SQLite works. Nobody stepped up to do that (and browser vendors weren't exactly foaming at the mouth to write their own SQL engine), so it died. You'd be better off writing a JS SQL engine that worked on top of IndexedDB these days, given that that's shipping in all major browsers.