It seems pretty obvious that shared online shopping via Instacart cannot be a long term solution for Publix because it minimizes their customer service strength with a generic interface and offering. My guess is they know that already and are working to correct it.
As for a "loyalty" program I really wish we would stop calling shopper history programs that because as others have pointed out, they really aren't. Knowing what your customers buy can fuel powerful customer service initiatives (in-store and online) helping a retailer tailor offerings to individuals and making it far easier for customers to buy from them.
Many shoppers move between chains effortlessly based on their weekly shopping needs and chain specials. Understanding what an individual shopper is likely to be looking for in a particular week and communicating to them when your store has that item featured can be a powerful motivator for them to shop that store during the week.
And if you do that every week, consistently matching your specials to their needs, it won't necessarily create loyalty but almost certainly will encourage them to adopt a default position of making sure they review your offerings each week. And that is a start.
From my experience, which healthcare providers people use has more to do with who is in their insurance network than anything else. Most plans pay significantly less (or not at all) for out of network care, which effectively eliminates any reality of choice for many people.
Asking customers what they want is never a bad idea, but they are already telling you everything you need to know when they shop. Not just what they buy, but when and why they buy the products they do, what discounts they require and so forth. Truth is, an individual shopper has interest in only a small fraction of the 30,000+ skus a store carries or even the 300 or so items currently being featured on "deal" so the biggest problem shoppers have is cutting through the clutter to find items relevant for them during a particular week. Using the full range of shopper data, not just the "did they buy this previously" metric so common, to determine which items in the weekly ad feature should be highlighted to each individual, actually delights shoppers and drives both incremental visits and sales each week.