John wrote:
A proper response to the student's statement would have been the simple reply that "technically there is a C# that is the enharmonic equivalent of the second semitone above C♭ on equal-tempered instruments."

From the practical point of view, a teacher who is teaching their student to read music should at least do this, so that if/when the student encounters a B# or C flat etc. in printed music, they understand how to play it. To start with, they may not need fully to understand WHY it's printed that way.

If a teacher says there is no such note, how do they deal with music that has that note in it?

A piece in the key of C sharp major has a B sharp in the time signature, and though it's unusual, it's valid.
More commonly, a piece in C sharp minor (only four sharps in the key signature) will have B sharps as accidentals.

Same goes for E sharp for music in F# major or minor.

C flat major is possible as a key too (seven flats) and contains an F flat, and G flat (a perfectly vlid and commonly found key) contains C flat as part of the scale.

If you are not teaching playing from music, you might as well not give the notes names at all...

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The issue of equal tempered vs. mean tone and the rest is a quite unrelated, though it does have some bearing on how notes are played. It doesn't only apply to flat/sharp/natural enharmonics either. In scales other than equal temperament, an E may be different in A major from an E in C major, for example.