No, Rayleigh scattering models the probability (and angle) of scattering as a function of wavelength and of the particle sizes. All wavelengths travel a long way but the path followed (scatter or nonscatter) varies.
Since space is mostly "empty", there's little scattering.
Beyond that, your understanding of stars is quite incomplete. THey do in fact have a variety of "colors," aka peak lambda outputs, depending on size, age, etc.
"White," as a color, is a perceived effect and can occur either when a bunch of wavelengths hit the retina simultaneously OR when the input intensity is extremely high. Then there's a third case: in very dim light (e.g. most stars during a very dark night), our color receptors don't respond and only the 'grayscale' cones in the retina respond.