The examples you are given are not natural, although maybe formally correct. Colloquially, you'll rather find
I am not a teacher, but a student
where but is adverbially linking student to the verb of the matrix clause We can readily see that I am not a student, [and] a teacher is ambiguous; different bracketing yields different ways to parse the negation; If anything, the intended negation is ungrammatical because the number of the singular subject disagrees with the two-headed object. Redundancy is thus required to disambiguate, e.g: I am not a student, not a teacher, which however is equally unlikely, and rather said using neither, nor. The base case I am a student, [and] a teacher [and a unicorn] is still ambiguous due to the semantics, where teacher and student are usually in opposition, so as well as might be used as the conjunction of choice to mark simultanity. Copula are regularly required between two indefinite nouns, except for adpositions (a teacher, a smart one).
The only thing that's redundant in your example is the repeated verb. If amn't is parsed as a lexeme, then I am not X, I am Y is anaphoric without redundancy. However, that's rarely the case, as shown in the frequent reversal I am a Y, not an X.
There's another reason to use the conjunction: *It is not the show it ..." rather expects "... used to be", a syntax that can be explicitly disambiguited with the conjunction that. It seems reasonable to assume the indefinite article would follow the same pattern of the definite article, for simplicity of the paradigm, even if a definite pronoun it cannot be in adposition to an indefinite noun (rather it needs that, which then functions as a determiner: It is not a show that is shown on TV). Moving out on a limb I guess we may still see It will not rain, it won't, which is prefectly redundant, that is seen all too frequently in reversed order in anaphoric questions, isn't it?