In many stores, they will swipe “a store card” for you at the register if you ask them. I believe I’ve even seen the option on some self-checkout stations. I’ve only had trouble with this once or twice. But it may be worth risking for the privacy (and monetary) savings.
This works for cases where just being a member gets a lower price, but at least some stores have digital coupons that need to be added to my account. This would not work with a generic card. It is definitely a good tip for stores where it does work though.