Roger Penrose in "The Road to Reality" says that the AFFINE SPACE has no cente,r i.e. no "origin". In contract, we can transform a linear affine space into a linear vector space by simply positing an ORIGIN i.e. a PREFERRED POINT.
Consider the ABSTRACT SPACE of ALL Lorentz Frames. For now only GLOBAL SPECIAL RELATIVITY of 1905 - forget the complication of 1916 GR for now.
A given Lorentz inertial frame is a "point" in this SPACE OF FRAMES.
Einstein's interpretation of SR is that this space of frames is an AFFINE SPACE.
Lorentz's interpretation is that this space of frames is a VECTOR SPACE with an origin.
This is an additional postulate that does not spoil any of the dynamical predictions that do not depend on a preferred frame. The "More is differenht" enmergent Macro-Quantum Theory of Vacuum provides the mechanism for the preferred "origin" in the space of frames, just like the preferred orientation in a finite region of ferromagnet. This is an empirical question.
Making the space of frames a vector space rather than an affine space is no big deal in terms of the set of battle-tested predictions of SR that are invariant under this narrowing down of the theory from affine space of frames to vector space of frames.