The rewards card is a much better model in my opinion because while it gives them quite a bit of data, it does provide some anonymity. I'm sure it is possible to reconstruct from that data who I am (i.e. convert it into direct PII like name and address), but that at least takes a lot more effort and processing than if they have my phone number.
Most people are ok giving up SOME privacy for the sake of convenience/cost savings. I doubt most people are truly willing to give up all privacy for said benefits once they understand what they are actually giving up.