>>I haven't been able to figure out why it is delivered as a browser extension, though
This came largely from alpha user feedback. Those folks told us that they wanted access to the question stream, but didn't want another login to manage to access Twitter data. So, we started with the extension which made it easy to be notified when there were new questions.
We are working on a web app as well as integration into some of the major Twitter clients and social media dashboards. Eventually we'll get to mobile (another highly requested platform). That way folks will have lots of options as to how they access our tools.