I believe this is the line that led to your question:
"Our analysis only applies for small deformations, for which the string is a linear medium, and we neglect the gravitational force on the string (which in any case is constant). "
note " ...small deformations for which the string is a linear medium..." i.e. the string is (conceptually) a spring that is not pushed past its elastic limit
it is the amplitude, not the frequency, that has to be small so you do not pass the elastic limit.