Back to Top Browser UI

I have been getting used to hitting the status bar on my iPhone to jump back to top... but I have moments of disappointment when I go looking for one in my browser. Question: Could it be in the default browser UI? Or should it be up to the website being viewed to design their own back-to-top button? Maybe something in between?

My idea is that these buttons (two versions shown) could slide down above the up-arrow in the browser UI when a scroll point has been reached. Thoughts and/or rebounds?