I'm a bit confused about the frequency response and the state of a system. Is it simply the ratio of the output of the zero state response to the input of the zero state response? Or does it include the zero input response? Any clarifications would be greatly welcomed.
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.
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.