Back when I was still trying to make sense of this, I theorised that the Reapers were what the Time Lords used to fix extreme cases. The logic being that the universe doesn't give a damn what happens to history, but the Time Lords do. Once the Time Lords were gone, there was no-one to control the Reapers, and presumably they chewed through the wire and got out.

The fact that history is changed anyway, because the car didn't hit Pete in exactly the same place and at exactly the same time, is what blows the whole thing to splinters. Nature abhorring a vacuum is one thing: nature deciding that oh well, it's technically a vacuum but we'll let it go this once is another. That kind of decision, the kind that says stopping the fall of Pompeii is changing history but saving one family who would otherwise have died isn't, requires some form of consciousness.