We had this for a brief moment in the form of Web SQL, but as a rule web standards require independent implementations to be developed, and all the implementors were using SQLite under the hood. The current alternative is IndexedDB - for better or worse.
> Is the browser environment becoming more dev-friendly with the adoption of WebAssembly?
Yes, but that doesn't have much to do with WebAssembly.
JavaScript as a language evolved, we have TypeScript with its flexible and expressive type system and bundling is becoming less of a pain with each consecutive framework re-written using ES2015 modules(and loaded via e.g. Snowpack https://www.snowpack.dev/).
WebAssembly is helpful when there's a need for some heavy lifting - e.g. in software codecs, but that's still a niche application.