Over the last decade, a growing number of quantum-gravity researchers has been looking for opportunities for the first ever experimental evidence of a Planck-length quantum property of spacetime. These studies are usually based on the analysis of some candidate indirect implications of spacetime quantization, such as a possible curvature of momentum space. Some recent proposals have raised hope that we might also gain direct experimental access to quantum properties of spacetime, by finding evidence of limitations to the measurability of the center-of-mass coordinates of some macroscopic bodies. However, I here observe that the arguments that originally led to speculating about spacetime quantization do not apply to the localization of the center of mass of a macroscopic body. And, I also analyze some popular formalizations of the notion of quantum spacetime, finding that when the quantization of spacetime is Planckian for the constituent particles, then for the center of mass of a composite macroscopic body the quantization of spacetime is much weaker than Planckian. These results suggest that the center-of-mass observables of macroscopic bodies should not provide good opportunities for uncovering quantum properties of spacetime. And, they also raise some conceptual challenges for theories of mechanics in quantum spacetime, in which, for example, free protons and free atoms should feel the effects of spacetime quantization differently.