Can predicative complements not be bare noun phrases in English? That is, are clauses such as “I am student” incorrect?

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?  

Solution 1:

The question seems to be, how does a non-native speaker determine whether a given noun is a "role" or if it has some titular sense (so that it can be used "bare")?

She became treasurer. [OK]

She became student. [not OK]

She became student of the month. [OK]

She became doctor. [not OK]

She became doctor to the king. [OK]

It was gateway to the rose garden. [not OK]

The city of St. Louis was Gateway to the West. [OK]

A role is typically occupied by one person or by a few, e.g. Assistant Vice President. Employee would not be considered a role because, in most contexts, an employee is one of many, even though we can imagine a small mom-and-pop candy store hiring only one employee. The operative condition: in most contexts, usually. In situations where there is usually only one such noun, there must be something special about the noun in order for it to be regarded as a role. (see the gateway example).

Student and doctor in the examples above are comparable to employee. They need something additional to identify them as particular student or doctor roles.

Solution 2:

"I am President of the USA." "I am Prime Minister of the UK."

The above are correct. There is only one of each at a particular time.

"I am student of physics."

This is normally considered to be incorrect because you are one of many students who study physics. Therefore you are 'one student' or 'a student' of physics.

"I am student of all things American."

This is allowable in English although usually considered formal. It focuses on your role rather than being a description of you.

Edit in response to comments

Examples

The teachers I teach are my students. They are also teachers – to their students. They are students of their students' learning, as I am student of their learning. They and I are also colleagues, fellow academics.

Using Experience for Learning edited by David Boud, 1993

Francis Blake, is student of theology in the Divinity School at Cambridge. Henry Bigelow, H. U. 1836, son of Lewis Bigelow, is student of medicine. John Healy Heywood, H. U. 1836, son of Levi Hey wood, is engaged in instruction

History of Worcester, Massachusetts - By William Lincoln

I was unable to find any recent examples of American usage.