The issue with WebSQL was, and still is, that it is basically impossible to standardize, as you wouldn't get multiple different implementations, but just one that all browsers use as a library. The main task of a browser is not to provide an SQL API client side, but to display interactive documents, so you won't find the bandwidth for browsers to build multiple SQL implementations, and the existing ones have quite big incompatibilities to each other.
It's a manufactured issue that is rather absurd on its face. If SQLite is so popular that literally everyone ships it and nobody even wants to talk about writing an alternative, then take it and standardize it as is, or at least some sensible subset of it. It's probably the most open and portable codebase of notable size in widespread use worldwide, so any portability concerns are extremely hypothetical, while the productivity losses from not having a portable relational store in the browser are very real.