I don't think it's really necessary to have a button do it as long as the Media Browser is towards the top in the in-game menu where it's easy to see. Really, there isn't going to be much point using the media browser when you're not playing a game. When you're in the game, the Media Browser will work very well with the rom you're currently playing.
To use it in-game, you would just press the right joystick down once, and selet "Media Browser" and all the different options will come up.
Otherwise, the Utilities is the best spot for it outside of the gameplay. The main menu is already full with options that are, in my opinion, more important for setting up the GUI and actually playing the games. The Media Browser is pretty much by definition a utility, so it makes sense to put it there.
And besides.... don't you know that people HATE button combos?
I can't tell you how many accidental game saves and screenshots I've made over the years.
haha... yeah I guess I have been known to take screen shots, add things to my favs, delete things....