Update Browser extension private repository support content/docs

Created by: erzhtor

Since we now support private repositories on Sourcegraph Cloud, we should update all content and docs. In particular, these are:

• Browser extension after install page
• https://docs.sourcegraph.com/integration/browser_extension
• https://github.com/sourcegraph/sourcegraph/issues/31920
• feel free to put other places if you find new...