Is Special relativity based on a fundamental flawed claim of inertial reference frames without acceleration which do not exist anywhere in space?
NO. Special Relativity is not based on any “claim” at all.
It is based on the observation that Maxwell’s Equations (and the consequent value of the speed of light) appear to hold with the same constants in every freely falling reference frame, and is only expected to be valid in situations satisfying the simplifying condition that gravitation has negligible effects on the quantities of interest.
That simplifying condition of course limits the domain in which SR applies; but the existence of locally inertial frames is readily apparent, and for the purpose of measurements over a sufficiently small range of space and time SR has no difficulty dealing with accelerated frames as well.