Consider a condensed system of electrons and protons in such quantity that their mass would be on the order of kilograms, and let this system be split into two lumps separated by some distance. If we only take into account electromagnetic, strong and weak interactions, can it appear that these two lumps of hydrogen would have an acceleration, which we'd normally describe as a gravitational force? In other words, is there a proof that the Standard Model doesn't predict any such "extra" acceleration however small it would be?
Neutron stars are mostly neutrons which are neutral. If gravity was a side effect of charge lumps as you described, then neutron stars would have much less gravity than observed/calculated.
There can be black holes without any charge as another example.