Timing Events are common in games, but a QTE is slang for a particular type of Timing Event, and they should not be equated with rhythm games or other Timing-based events. In fact, they should probably be more properly called QTME, or Quick Time Movie Events. QTMEs exist because, as the non-interactive movies were getting longer, many players were getting more and more bored while just sitting there watching movies while they were supposed to be playing a game. QTMEs add some amount of interaction back into the movies.
But that has always brought up several issues:
1) QTMEs generally amount to "press XYZ to continue", instead of "press spacebar to continue". While that does allow the player to input commands during a movie, he is not really interacting with the game. Such a QTME has barely any functional interactivity, since the inputted commands have little to do with what is happening on screen. And that is not real gameplay.
2) Why break up gameplay with a movie in the first place? It's a game, not a movie. There should only be a movie if it's something that can't be told even remotely as well in any other way.
3) Okay, you've decided to break up gameplay with a movie. So, why break up that movie and try to frankenstein gameplay back in? After all, you already broke the gameplay in order to have your oh-so-special movie, but now you're telling the player he can't concentrate on your oh-so-special-movie, because he has to click something to continue within a certain time-frame, thus pulling his attention to the key-presses and the time frame instead of your movie. And thus, you lose out on both worlds, having dropped both good gameplay and good story in exchange for stupid gameplay.
There were once a number of games released based around QTMEs. They occurred during the FMV days, and include such wondrous titles as Critical Path. None of them are remembered today. Okay, I will allow that it is theoretically possible to make a fun QTME game. Lots of things are possible in theory. But QTMEs place such a limit on gameplay that the chance is laughably remote. Rather, you could make a fun semi-interactive movie out of them, but there is scant chance of getting a good game (or even a game at all) out of that semi-interactive movie.