There is the *minor* issue of having an infinitesimally fine resolution of time. Measuring one period also assumes that the next is exactly the same length. Very unlikely to be true in the real world.
You can actually do far better by sampling the waveform and doing an L1 (least summed absolute error) fit. That avoids the need for a super fast timer clock. But it does require a non-trivial amount of computation. TANSTAFL
Shannon worked out how much data is needed for a given amount of information at a certain level noise (error). While Shannon was focused on the data capacity of a communication channel, his work has broad application in many areas. For example, mathematically the phase can be derived from the amplitude spectrum alone on physical grounds. There is, however, a seriously non-trivial issue of how much information is needed. A VNA can measure phase at a single frequency, but you cannot derive the phase from a single amplitude measurement. I don't yet know how many amplitude measurements are needed to derive the phase to a given accuracy, but I'm quite sure that Shannon's work applies.