I never mentioned time dilation as an observable effect of gravitational waves though (which is also not the case for a stationary observer ). You are right in that it is observer dependent, I am implicitly referring to first order perturbation theory to minkowski spacetime for a stationary observer on earth which is just the simplest model assumption for the purpose of explaining and detecting gravitational waves.

If we are talking about atomic clocks you would have to solve a Schrödinger equation in explicitly time dependent curved spacetime which I would not want to make an uninformed guess about ^^

And if you measure e.g. the time it takes for light to travel an infinitesimal distance you still measure coordinate time i.e. proper time from the perspective of a stationary observer in only spacially perturbed minkowski space.