I don't think that's the case.
At least where I'm at, almost none of my Amazon orders come through fedex or UPS now - everything is delivered by an Amazon driver. And they have an incredible amount of Amazon vans running around. I went to lunch the other day and counted 3 within a mile of my house.
That does make me wonder how they optimize routes. That and the fact that I had 4 different drivers leave a package at my house the other day.