IndexedDB is fine for storing JSON objects, etc. but a relational database with SQL query syntax, indexes, etc. more powerful and means less code to write. With IndexedDB one has to reinvent the wheel to just get basic query features.
WebSQL is not deprecation, the W3C Working Group Note actually says:
'This specification is no longer in active maintenance
and the Web Applications Working Group does not intend to
maintain it further'.
WebSQL is only available in Webkit based Browsers (Safari, Chrome) which means most mobile browsers.
As SQLite is in public domain, no company would "loose their face" if they choose to use it. They could fork off SQLite and change the SQL query syntax (parser) to whatever the W3C finds suitable. https://www.sqlite.orgMozilla Firefox and FirefoxOS both already ship SQLite for years and can be accessed by its internal JavaScript API. And several Microsoft products already use it anyway (e.g. Forza Xbox games). Microsoft has of course also various other SQL database libraries like MS Access JetRed, MS Outlook JetBlue and SQL Express.
We had a discussion about it recently: https://news.ycombinator.com/item?id=7574754
The new hip things is "NewSQL" (http://en.wikipedia.org/wiki/NewSQL ). For example Facebook, Google Ads, etc. are powered by MySQL's InnoDB database engine. I would go as far as count SQLite to this group.
We would need a movement to convince Mozilla to finally add WebSQL to Firefox and FirefoxOS.