We study generalized random fields which arise as operator rescaling limits of spatial configurations of uniformly scattered weighted random balls as the mean radius of the balls tends to 0 or infinity. Assuming that the radius distribution has a power law behavior, we prove that the centered and renormalized random balls field admits an $\alpha$-stable limit with strong spatial dependence, according to the attraction domain of the weights. In particular, our approach provides a unified framework to obtain some operator-scaling $\alpha$-stable random fields, generalizing the isotropic self-similar case investigated recently in the literature.