I don't even doubt that they'd do this. Imagine trying to read something while some advertiser constantly tries to grab your attention. Is there a single place on this planet that's safe from advertisers?
Most libraries are relatively lax on the proof because they see it as a public service. But they don't have to be, and you would expect that a public library operated by a for-profit company will be far more strict than others.
I imagine a for-profit library would be largely frozen out of our interlibrary loan system and all the reciprocity involved in membership, but... maybe not? I don't know, but I suspect that this model of deprofessionalisation is going to be limited to libraries in small towns and municipalities.
All the books in my public library have RFID already to make them faster to check in and check out. I'm also pretty sure there's been a record of every book you've checked in and out of the library even predating that. Glad they don't already do this.
"The Library does not maintain a history of what a borrower has previously checked out once books and materials are returned on time."