What they did was increase the "regular" price of retail goods above what competition would charge, apply a "discount" if you have a Kroger card, and then you get this "cheaper" price. They even put how much you would have "saved" on your receipt if you didn't have a card but that's vs inflated prices others don't charge. At one point, their "regular" price for milk was at convenience store levels with sale price just over Walmart. These fake discounts should be considered straight-up fraud. That's not to say they don't have plenty of real ones but you'd have to look carefully to know. There's yellow tags everywhere. So, about all of it is cheaper than Walmart or a similar competitor at that moment, right? Nope.
The other thing they did with Kroger card was price discrimination. The regulations apparently were about the price of items rather than what people paid for them. These are technically two different things. The way that works is they learn that certain people will keep buying specific things. They use this data from their card profile to deny them coupons or other discounts while they give them to other people. They've got a lot of profiling tricks for adding or denying discounts. I'd say more consumers take a hit than benefit since there were previously coupons for all of them in the papers or what they mailed out. They're also mailing out fewer coupons with lower amounts, too, now that they've rolled out digital coupons tied to profile data from card.
So, many of the sales are fake. The line about what people would've saved versus Walmart is fake. Most of the coupons are for brand-name items that already overcharge you anyway. The good generics are cheaper much of the time. The gas discount and recall notices are about the best benefits of the Kroger card. The best coupons they gave me discount per order, department, etc letting me spend on whatever I wanted. Those could probably be delivered based on survey data without tracking.
Summary: the Kroger card is mostly a fraud designed to trick people into thinking they're getting discounts, give them coupons for more expensive stuff, give them less coupons that would make things cheaper, and otherwise make profit on surveillance of them. There's benefit but most selling points are layers of lies. This is why I prefer either no loyalty cards or them limited to stuff having nothing to do with pricing. Preferably by law to increase competitiveness and reduce risk to consumers.