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It's more like, my mother goes to the shop, the store person looks up the inventory, doesn't find the item, she asks them to order it, etc.

Prime usually delivers within a day or two, unless heavy snow closes off the only semi-paved road leading up to her place :-)

No, I think it's a different sense of "faster" -- it delivers with less than an hour's effort on the part of the customer.

At least one flawed assumption you make here is that the drive can be executed immediately.