People who say "they should just buy the DVD" are imagining that the library can just pay $20 or whatever, and then offer it to as many users as want to watch?
I don't think it works that way for libraries. Maybe someone who knows for sure can tell us? I'm guessing that they get the right to loan out the DVD physically but not electronically.
I ran the Cinema Club at Google for 10 years, and we made our lawyers happy by always paying "public performance rights" (PPRs) for all our flicks. When you buy a DVD, you get a license for "home viewing." That does not allow you to invite members of the public in to watch. You cannot show it in your bar or restaurant or at a place of business. Maybe you, Fred Hacker, can get away with it, but a large multinational company cannot.
When I negotiated PPRs with a film distributor, they always asked these questions:
1) Is the film showing advertised outside your company?
2) Is attendance limited to employees and their guests?
3) How big is the room? (or maybe it was "what's the average attendance?")
I didn't look into what public libraries pay for their movies, but I'm pretty sure it's more than you would pay to just buy the DVD.