Just a web browser is definitely not part of the core OS. It is just an application, no matter how important.
This is why we went with WebKit, as I said. The project goes further than just the web browser, bringing the availablility of HTML rendering into any application.
Our vision for Haiku is not just doing things in the web browser, but rather writing native apps that interact with the web. You can see this with apps like Radio, the work in progress calendar application, the Weather app, or HaikuToDo. These are small examples of what we're aiming at, and being able to render HTML directly inside native applications will likely play a role in that.
As a result, the project is more ambitious than just getting the web browser running, and most efforts are concentrated on the engine rather than the "chrome" (that is how they call the user interface - and where Google browser's name comes from). This is why WebPositive is a bit underwhelming as a web browser - it is there only to showcase the engine under it. However, it would be great if someone could step up and improve it, and it is also great that there are alternatives like Qupzilla until we get it ready. But, the main focus should still be on the native apps and integration with them, not just getting some random web browser to run.