But you run into the same issues mostly because Kroger's delivery service is "just as 3rd-party" as Instacart. It's still an employee going through your store, grabbing stuff off the shelves and delivering them to you. There's basically zero first-party advantage.

Yeah, this is exactly what it was. We used a Kroger store because they had a much better selection than Safeway and in the end didn't really get what we wanted. At the time I had guessed it was because the 3rd-party picker didn't know where some of the items were and didn't have time to go looking for them.

At least the in-house grocers are more familiar with the layout and product offerings. From a process perspective, couldn't supermarkets process the delivery orders during stocking?

Having been warned about Instacart I made a point of doing just that, only to have the grocery store run the order through them anyway. The whole experience felt fraudulent, from the bait-and-switch between service providers to the bait-and-switch from carefully chosen sale items to expensive and inappropriate "substitutes". Reportedly at least Instacart proper has a feedback mechanism. With the white labeled service you're out of luck.