Certain excitations, especially ones of long-range charge transfer character, are poorly described by time-dependent density functional theory (TDDFT) when typical (semi-)local functionals are used. A proper description of these excitations would require an exchange-correlation response differing substantially from the usual (semi-) local one. It has recently been shown that functionals of the generalized gradient approximation (GGA) type can yield unusual potentials, mimicking features of the exact exchange derivative discontinuity and showing divergences on orbital nodal surfaces. We here investigate whether these unusual potential properties translate into beneficial response properties. Using the Sternheimer formalism we closely investigate the response obtained with the 2013 exchange approximation by Armiento and Kummel (AK13) and the 1988 exchange approximation by Becke (B88), both of which show divergences on orbital nodal planes. Numerical calculations for Na-2 as well as analytical and numerical calculations for the hydrogen atom show that the response of AK13 behaves qualitatively different from usual semi-local functionals. However, the AK13 functional leads to fundamental instabilities in the asymptotic region that prevent its practical application in TDDFT. Our findings may help the development of future improved functionals. They also corroborate that the frequency-dependent Sternheimer formalism is excellently suited for running and analyzing TDDFT calculations.