At least in Europe, you need a suitable legal basis for processing under the GDPR, at least in theory (we do know however that GDPR enforcement is significantly lacking so this might be just a false sense of security as there are no consequences to breaching the regulation).
Signing up for a loyalty card would give them that legal basis, where as merely using your payment card wouldn't, so they theoretically shouldn't be stalking you. In practice, not giving them your name/contact details also means there's no chance of spam arriving from them. There's was never a moment I said "I wish my supermarket would contact me out of the blue for any reason" and I'd like to keep it that way.
There is exactly one reason why I wish a supermarket would contact me out of the blue: a recall, since they would in theory know what I purchased if I used the card, so no need to first hopefully hear about the recall, then try to figure out if it applies to me.
That said, I've never heard about that as a benefit from a market in the US, just discounts on food items (or put differently, upcharges if you don't use a card).
Not that we have GDPR-level protections here either.
Signing up for a loyalty card would give them that legal basis, where as merely using your payment card wouldn't, so they theoretically shouldn't be stalking you. In practice, not giving them your name/contact details also means there's no chance of spam arriving from them. There's was never a moment I said "I wish my supermarket would contact me out of the blue for any reason" and I'd like to keep it that way.