Yes, I believe that will be one of the results of setting the browser in the constructor when the repository URL is a recognized URL.
The constructor would continue to be called with the same arguments it has today. Inside the constructor, there would be a call to decide if the repository URL is a recognized URL. That call would be an extension point which is used by the git plugin to register the recognized URLs. Since it would be an extension point, other plugins could add their own recognized URLs without requiring changes to the git plugin.
There is a page that describes how to define a new extension point. Refer to the current Jenkins extension points for possible reference implementations. The RepositoryBrowser is an extension point that is implemented in the git plugin repository browsers. I've never created a new extension point, so this will be a learning experience for me as well.
I want to contribute on this issue. From where should I start ?