I can't speak to IOS but it will on android devices (with some limitations https://developer.android.com/about/versions/oreo/background...) If you give an app permission to access nearby wifi devices it can track you that way as well. If you give an app permission to use bluetooth it can track your location using that too.

> They'll profile you based on your credit card info whether you use the app or not.

They might, but they'll have a lot less data to connect with your purchase history. You consent to a lot more data collection by installing their app than you do just by walking into their store and buying something,

> The mom-and-pop places that people who make a fuss about these kind of business practices tend to love do all the things that those people claim to hate - track their customers, try to figure out how much money they have, offer special treatment to particular customers - so I've never quite got what exactly people are objecting to.

No one wants to be tracked by mom-and-pop places either. All this data collection means that you aren't getting a discount, you're actually being screwed over and the data they collect will continue to be used against you for the rest of your life in any way that they think will let them take more of your money or would otherwise benefit them. They aren't collecting that data and paying for the collecting/storage/use of that data for your benefit.

"Special treatment" doesn't mean what you think it does. This kind of surveillance leads to discriminatory pricing (see for example https://www.semanticscholar.org/paper/A-special-price-just-f... and wendy's recent attempt at surge pricing) which companies are desperate to get consumers to accept because it stands to make them much much richer at your expense. Don't fall for this! I promise that you won't feel like you're getting "special treatment" if they get their way.