Does the frequency response assume the zero input response to be 0?
A frequency response describes a linear time-invariant (LTI) system. It is the Fourier transform of the system's impulse response, and only LTI systems are fully characterized by their impulse response.1

A system with non-zero initial conditions is not LTI because its output has a component which only depends on the initial conditions and which is independent of the input signal. Consequently, a system with non-zero initial conditions cannot fully be described by a frequency response. If you assume zero initial conditions, then such a system (if otherwise LTI) can be described by a frequency response.

So to answer your question, a system can indeed only be fully described by a frequency response if its zero-input response (ZIR) is zero, because if its ZIR were not zero it wouldn't be an LTI system.

Also take a look at these two answers to related questions: answer 1, answer 2.


1Also linear time-varying (LTV) systems can be characterized by an impulse response, but it is a two-dimensional function, and LTV systems don't have a frequency response in the conventional sense.