Not necessarily. The easy way to implement this in-person would be to give customer-specific coupons. You could get an email that says "use your loyalty card at checkout and get $2 off eggs". Then you just give everyone a different discount and only the privacy-minded folks end up paying the (inflated) sticker price.

A significantly more complex hypothetical that I don't think anyone is doing yet: With digital price tags and customer tracking you could show different prices to different customers in-person. For example, when Alice goes to the eggs it could say $2 and when Bob goes to the eggs it could say $4. Then you just need to track the customer to the register to make sure you give them the price that was displayed. I believe the amazon "go" stores were doing the whole customer tracking thing so we already have the necessary tech demonstrated in real stores.

My mainstream supermarket in Nordics has for the longest time given individual coupons in their loyalty program.

There are three tiers of discount. There are general special offers, advertised widely, to get people through the door. Then there are members only discounts, advertised in store, to get people to be members. Then there are mailed to members personal discounts that are uncannily accurate for what is running out to get you to go shopping now instead of later. These days the hand scanners and apps also tell you what coupons you have so you don’t need mail but junk mail is still regular.

TBH as a shopper it works great.

Sometimes, you may not have an Aldi near you. Also, many people may not have the luxury of going to grocery stores because they have to work in an office for 8–9 hours.