Why does Islam not recognize a no fault divorce?

If you mean by no fault divorce where both parties have agreed to go their separate ways and simply file and are done.

A no fault divorce under the legal system simply means that you're not trying to say that this person committed infidelity or there was abuse or anything like that.

You're simply saying, I don't want to be with this person anymore. And that's fine. And that happens Islamically under talaq as well as khula. For example, the wife of Thabit ibn Qays Shammas. Who was one of the instances of khula during the life of the Prophet, peace be upon him. She went to him and she said, Oh Messenger of Allah, I don't want to be with him anymore.

Okay, do you find a problem? She said, No, I don't find any problem in his character or his deen. But when I look at him, I want to spit in his face.

She had completely grown apart from him and because of that the Prophet, peace be upon him, said okay, here's the khula process.

Similarly for a man even in a no fault divorce, during the waiting periods you're still going to be living like you're living.

So, the no fault divorce does not have a benefit over the talaaq and khula processes because there's always going to be one party who files, and they will be the breaching party, and there will always be things that have to be sorted out, which will be the issues of maintenance and property, just as they would in an Islamic court.