The Library is run by people who believe themselves to be guardians of all knowledge and as such have superiority complexes. When you have access to "all the knowledge in the multiverse", you clearly are also very powerful (not to mention that they seek out powerful magicians to join their ranks-ie. Penny.) Some Librarians are more corrupt than others but even Zelda, who is arguably the purest, often meddles in affairs because of her perceived superiority. The Librarians see anyone who is not involved with the Library as a threat to their goal to protect knowledge. They seek to collect and protect books, but the only knowledge they actually create is the books about people's lives, as shown in season 4. **
With all this in mind, it seems like the only smart choice of the Librarians would be to control all magic if given the once-in-a-universe chance. Obviously from the protagonists' point of view it is evil and unnecessary but consider being an essentially immortal being dedicated to the collection and storage of information. Being able to control people you see as a threat (hedges, Fillorians, magicians like the Beast, or upstart Brakebills students) by cutting off their magic is a safe and efficient answer. For the more cynical Librarians, it gave them a chokehold of authority over one of the most important magical places on Earth--Brakebills--and allowed them to dictate educational policy for the up-and-coming magicians who could potentially be either threats, cardholders, or even Librarians.
In season 4, after escaping from her cell, Alice finds Christopher Plover hiding out in the library. Together they steal their books along with Alice's friends' and rewrite some of the ending so the Library doesn't know where they're hiding.