Update Browser API migration to async note

[andrewslotin]

What?

Starting from v0.52.0 Browser API is fully async. This PR updates the outdated note to mention this and refer users to the migration guide.

Checklist

• [x] I have used a meaningful title for the PR.
• [x] I have described the changes I've made in the "What?" section above.
• [x] I have performed a self-review of my changes.
• [x] I have run the npm start command locally and verified that the changes look good.
• [x] I have made my changes in the docs/sources/next folder of the documentation.
• [x] I have reflected my changes in the docs/sources/v{most_recent_release} folder of the documentation.