Among the scalar-tensor modified theories of gravity, degenerate higher-order scalar-tensor (DHOST) models could play a special role for dark energy while being consistent with current observations, notably those constraining the speed of gravitational waves. Schwarzschild–de Sitter black holes were shown to be exact solutions of a particular subclass of quadratic DHOST theories, while carrying a nontrivial scalar profile that linearly evolves in time and hence potentially providing exciting new phenomenological windows to explore this model. We investigate the physical perturbations about such black holes and find that the odd-parity tensor perturbations behave in a way indistinguishable to general relativity. On the other hand, the effective metric for the (even-parity) scalar perturbations is singular, indicating that those exact black hole solutions are infinitely strongly coupled and cannot be trusted within the regime of validity of the DHOST effective field theory. We show how this strong coupling result is generalizable to a whole class of solutions with arbitrary manifolds both for DHOST and Horndeski.