SQLite has a _lot_ of great features. Many of them only introduced in recent versions, and many of them are optional extensions enabled at compile-time. There is no feasible way to standardize this to get the same functionality across all browsers.
IndexedDB may not be great to work with (at least not without helper libraries), but at least the API surface is small, and much easier to ensure you get the same behaviour in all browsers.
I'd rather advocate for a proper block-level filesystem API that could be used as backing storage for a WASM SQLite library.