Your argument is still not making much sense. I don't see how the power efficiency of the APIs is relevant. The point is that they currently allow tons of applications (E.g. Candy Crush) that drains the battery. What's the reason for making the browser a special case? It's just rendering to the screen like any other application.

One argument I could see being made is that the browser is used more than others and thus warrants tighter control. But even that is weak. People are draining their batteries all day long playing games as it is.