Why not? Your taxes pay for the library, so you own a share of their media. If you really want to be honest, check their online catalog and make sure the title is available for checkout before you play it.
However, "own a share of their media" means what, that I can have a legit claim to 30s worth of the content? How is that calculated? Divide the number of tax payers in the city by the number of seconds of each bit of content to see how much of it I "own"? You're really bending the concept of tax payer funding here. If that's how you want to go through life, then you do you
My only major critique of it is the library doesn’t know you wanted to borrow then, and usage metrics are very important for them when it comes to deciding what to purchase. But I get what he’s suggesting, it makes sense even if it’s imperfect (I personally wouldn’t do it).