It's not so much meant to be a minigame on its own as it is just a way of making a cutscene a little more interesting and immersive. When used well, it works. What I don't like is when they put in the generic "press the button on-screen as soon as it comes up" thing. It was alright for RE4, one of the first to use it, but it's definitely stale now. What's better is some method of control that closely relates to what your character is doing.
For instance, Modern Warfare 2: To climb an ice ledge, press the left trigger (move the left hook forward) then holding it, press the right trigger (move the right one forward) release, and press left again.
That sort of thing gives a very light feeling of being there, so it's good.