But then they have your phone number and can probably extrapolate your address from that

The cynic in me feels like they wouldn’t care. The deal they have with your phone provider allows them to identify you by the phone you signed up with. The deal they have with Amazon facial recognition let’s them identify you with the cameras you walked past every time you go in the store. The deal they have with the state let’s let track where your license plate was seen this month. And of course they have what they want which is the card in your pocket giving them all of your purchases in one easy column.

/s.....hopefully

Actually that's exactly what they and Dunnhumby do.

I certainly wouldn't install an app to go shopping at a specific store to obtain specific advantages.

No company deserves loyalty through a scheme like this. Only through decent customer service and reasonable pricing.

As for shopping elsewhere, yes. I use Asda now which does not require a loyalty card to get the discount prices and is generally cheaper and better stocked despite being smaller here.