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

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.

"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.