Good question.
First to address the main point: why not cache the relevant bits in some kind of local storage. SQLSync plans on doing this, specifically using OPFS for performance (but will have fallbacks to localstorage if needed).
Second to address the question of why not use built in kv stores or browser side databases. One answer is another question: how do you solve sync?
One approach is using a data model that encodes conflict handling directly, like CRDTs. This approach is easier to put into general kv stores, as syncing requires simply exchanging messages in any order. I find this solution is well suited to unstructured collaboration like text editing, but makes it harder to coordinate centralised changes to the data. Centralised changes are nice when you start introducing authentication, compaction, and upgrades.
Another approach is doing something similar to how Git Rebase works. The idea is to let the application state and server state diverge, and then provide an efficient means for the app to periodically reset to the latest server state and replay any unacked mutations. This approach requires the ability to re-run mutations efficiently as well as efficiently track multiple diverging versions of the database state. It's certainly possible to build this model on top of local storage.
For SQLSync, I found that by controlling the entirety of SQLite and the underlying storage layer I was able to create a solution that works across platforms and offers a fairly consistent performance profile. The same solution runs in native apps, browser sessions (main thread or workers), and on serverless platforms. One of my goals is to follow the lead of SQLite and keep my solution fairly agnostic to the platform (while providing the requisite hooks for things like durable storage).