To a large extent store chains already have them, that's the (few) logistics/distribution centres out of which they supply the (many) stores. The store shelf is essentially just a short-term cache.

Are the preexisting logistics centers designed around picking up single items off the shelves? I would have expected them to be set up around bigger blocks of product.

The preexisting logistics centers are designed to take a pallet of item X and allocate different amounts of it to different boxes each going to a different smallish shop; doing so in an multi-hour cycle (2-3 times a day?) to ensure that every type of product is distributed along these boxes. During this process they also put part of the product not in "delivery truck going to shop A" but to "delivery truck going to home deliveries". It does need some ripping of bigger blocks of procuct and repackaging to bags, but it happens at the same time as they're ripping even bigger blocks of product for supplying stores and shelves.

But the essential difference is the "push vs pull" - instead of one delivery pulling items from everywhere (which is a new process that grocery chains did not have) you have the supply process pushing items to many delivery targets, which is a process that grocery chains already have and have optimized.