In Chapter 4 of the book A Student’s Introduction to English Grammar, written by Rodney Huddleston of the University of Queensland and Geoffrey K. Pullum of the University of Edinburgh and published by Cambridge University Press in 2005, it was stated that
A bare role NP [noun phrase] is a singular NP that is ‘bare’ in the sense of lacking the determiner which would elsewhere be required, and that denotes some kind of role, office, or position. A PC [predicative complement] can have the form of a bare role NP, but an O [object] can’t:
i a. She became the treasurer. b. She knew the treasurer.
ii a. She became treasurer. b. *She knew treasurer. [ungrammatical]
On the contrary, there is a question (Is this proper English: "I am student"?) I found here on StackExchange on the similar topic, in which the OP was wildly discredited, and the question was closed for lack of research. I have the same problem, and here I am posting my research, which is also the origin of my question.
Before I read this book, I also believed that it was wrong to say things like “I am student” without any determiner. However, it became clear to me that apparently in numerous Indo-European languages, clauses like je suis étudiant (French) and Ich bin Student (German) are completely grammatical, whereas somehow the version with the determiner (je suis un étudiant or Ich bin ein Student) is less preferred, if not completely ungrammatical. So, why, for some curious reasons, is it definitely deprecated, if not ungrammatical, to say things like “I am student” in English?