The Prime service only works because Amazon is able to negotiate shipping prices due to their volume of pickup. Maybe if you were getting a high volume of delivery everyday, then shipping companies would consider it.
Also because they basically have warehouses and distribution centers in one. Instead of shipping your box to a distribution center and from there to your home, they just put it in the truck that delivers it to your door. At least that's what I've seen in Europe, but wouldn't be surprized that that is what they do in the US, too. I think they have direct 'peering' arrangements with DHL and others, in that the truck actually starts at Amazon or goes there the first thing in the morning.
I would love to see "independent" fulfilment centers. E.g. I order at my favorite company a few cities away, and my local fulfilment center (the former huge amazon warehose) processes it, and I get it next day. I could imagine a shipping flat, for all packages that go through "my" center. Note, I am not thinking "mom and pop" centers, I am imagining local joint ventures beween logistics/delivery companies and these huge warehouses/goods transportation centers. But still a few orders of magnitude smaller than Amazon. (For the Germans: all my Amazon packages go through Bad Hersfeld. Imagine that logistics center would be it's own company, maybe backed by DHL / Schenker / Trans-o-flex / ... I believe it has a critical size to be viable.)
Unfortunately, the free market is not going to bring me that...
This is kind of the idea of Fullfilment by Amazon (FBA). You send your stuff to Amazon in bulk, they keep it in their warehouse, and when you sell an item, tell them where to deliver it to. You can use it even if you don't sell on Amazon.
Prime works because not everyone makes frequent purchases and they get a flat $99 a year. People with prime are going to shop at amazon first because of the sunk cost of prime.