Yes, browsers should take responsibility for doing it without asking the web devs to handle it, with the possibility of messing it up.

Operating systems have spent years dealing with power saving policies and application requests for keeping the display/disk/music/connection on. After about 10 years we are now reaching a point where all this have acceptable defaults and decent GUIs to change the settings to suit one's needs. Now all this will have to be redone again inside the browser.

Shortly people will complain that the browser did not let the monitor sleep at the end of a video/movie. Or, in the future, that the browser did not prevent the computer from going to sleep while the music was playing AND that the browser let the computer to sleep while the music was playing.