Because you're not supposed to dodge.

Are you referring to dodging from champion select?

I guess it wouldn't matter what you're referring to though, because you are not supposed to dodge. It makes the game experience worse for everyone by making them wait longer to play.

Riot isn't going to develop a feature just to enable people to do something they're not supposed to be doing in the first place.

Because riot does not want people dodging which is why there is a penalty for doing so. Why would they go out of their way to make doing something easier that they dont want in the game? Because if there was a built in option to dodge a lot more people would do so on top of the already annoying amount of people that do simply due to them not getting the role they wanted

There's no feature of "dodging the game without closing the client", because you're not supposed to dodge, that's pretty obvious.

What does it matter? You need to wait out the timer anyway

Actually, there are legitimate reasons to dodge a game so thats why the penalties arent too harash