That wouldn't be cross-platform. Browsers couldn't even ship SQL because it would inevitably tie them to sqlite, specifically, forever. They definitely can't ship something that requires a whole Linux kernel.
But you are changing the goalposts. Browser's couldn't ship SQL (the JavaScript feature called WebSQL) because it would tie the web ecosystem to the future of sqlite, which is a specific cathedral-type project.
Nonsense.
Chrome store their sqlite db in C:\Users\%USERNAME%\AppData\Local\Google\Chrome\User Data\Default\databases
And Firefox:
> Why Firefox Uses SQLite
> Cross-Platform Compatibility: Works seamlessly across all platforms Firefox supports.
https://www.w3resource.com/sqlite/snippets/firefox-sqlite-gu...