If it's out of stock to the picker, it's out of stock to the customer, regardless of whether the back room has more. The robot will report to the back room when a restock is needed, but if restock hasn't happened yet, the item is not saleable.

Webvan did pick-from-controlled-inventory 20 years ago. Worked fine, cost too much because they didn't have enough market penetration. They had 3% in 30 cities, and they needed 30% in 3 cities. Amazon is getting into that, cautiously.

There's quite a bit of just-in-time delivery for grocers. Years ago when I worked at a large supermarket, we only kept stock of items that would reasonably need restocked within the next day or so. Things like yogurt, milk, eggs, soda, chips, etc. The majority of the items on the "inner" items (ie not staples) had 0 stock in the back room.