I think the current state is fine. You ship your WASM-blob of SQLite, which has the exact bug-compatible version of SQLite that you've tested your app against. The browsers are not burdened with maintaining a huge API surface that can "break the web".

Otherwise you'd have to deal with different versions of SQLite in different browsers, most likely outdated, with many options turned off. SQLite is full of quirks and gotchas, so it's safest to ship your version in your app.