The problem is that they can't control open source drivers. They could, however, release a printer that ran on proprietary closed source drivers. But they'd have to spend money on developers to maintain that code whereas right now, drivers are more or less stable and developed for free.
What they could do is require the use of HP printer paper, with embedded RFID or watermarks that would be readable by HP printers. I'm honestly surprised they haven't gone down this road.