Amazon already (sort of) do this in the UK for some of their deliveries, with deliveries often coming from an arm of their business called "Amazon Logistics"
I don't think it's a pure courier company though, I think they've just paid a bunch of different couriers to deliver parcels under the Amazon brand. For example if you are out, they'll leave a card with Amazon branding over it
I've recently had a few of my Prime deliveries come via "Amazon Logistics" and it's lacking compared to DPD who used to do most of them.
Amazon Logistics has no depots (that I can see - they don't tell you who the courier is), no real time tracking, no delivery windows... If this is their attempt at doing it on their own it's a terrible start.
I don't think it's a pure courier company though, I think they've just paid a bunch of different couriers to deliver parcels under the Amazon brand. For example if you are out, they'll leave a card with Amazon branding over it