We give a novel characterization of the centered model in regularity structures which persists for rough drivers even as a mollification fades away. We present our result for a class of quasilinear equations driven by noise, however we believe that the method is robust and applies to a much broader class of subcritical equations. Furthermore, we prove that a convergent sequence of noise ensembles, satisfying uniformly a spectral gap assumption, implies the corresponding convergence of the associated models. Combined with the characterization, this establishes a universality-type result.