The whole point is that the hotel would not even get a copy, the machine would just send hashes around and the hotel would only get an anonymous transaction ID to store. It would probably be even more secure than what you have today at the airport.
If you think photocopies kept in some folder accessible to anyone working in the hotel, with a promise to delete it at some point, is "secure" in any way, I don't know what to tell you.
Good lord!
Rather a paper data breach exposed to a few hotel employees than eletronic data exposed to the entire planet!