Does Zero Padding Work as Advertised?
Zero-padding data for a longer FFT is equivalent to interpolation by a (periodic) Sinc kernel.  Interpolation by a (periodic) Sinc kernel can reconstruct points between samples of a signal that was strictly bandlimited (to below the Nyquist frequency) prior to sampling.  See: https://ccrma.stanford.edu/~jos/resample/Theory_Ideal_Bandlimited_Interpolation.html .  This is a property of bandlimited signals (e.g. a linear-phase "brick-wall" bandlimiting FIR filter looks a lot like a Sinc function).

A lot of real-world data can be made close enough to bandlimited to meet existing or needed S/N criteria, and thus a zero padded FFT is "close enough" to "work" as needed.

It is not necessary to implicitly assume the repetition of a window of FFT data for all uses of an FFT.  The phase vocoder and fast convolution algorithms require the assumption that the data is windowed and not necessarily periodic in FFT aperture width, otherwise one could end up with a contradiction regarding the data samples across overlapped or adjacent windows.