Can a librarian handle the job/role of a document controller better?

Yes, a librarian can handle the role of Document managment.

Yes, they can handle much better this job since this is what they are known for, to properly maintain the library in a way it will be convenient for students and teachers what books they are looking for and using for.

Yes, of course.
A Librarian in his profession, deals with documents, be it print or non-print.
Each and every document, even if it is small, has its own value and should be reached to its particular user.
This is the ground on which librarians work.

Yes, Librarian can handle document controller role coz he is the one who segregates the book in the form of document and avoids duplication and manage well with proper title.
 

Yes, a Librarian can handle the job /role of a Document Controller, but I cannot considered the word 'better' being a Librarian.
In my opinion, as long as the person has the knowledge to learn the systematic procedures in Documentation.
Knowledge in computer software, creating data sheets and procedures to make the archiving efficiently.
Most importantly, they should have the knowledge or the experience in such field, ie.
the type of business, like construction.
Being a Librarian, he has no knowledge of what is happening in the construction business, it is entirely different.
In this kind of job as a Document Controller, it is itself, learn by experience by the person who has the patience to learn the proper and effective procedure., its either, manual or electronic.

A librarian isn't professionally a document controller. Library science & document controlling are different tasks / jobs. But a librarian should have basic knowledge of document controlling as it will be helpfull during his responsbilites in Library.