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The push gravity model assumes an "ether" of randomly moving particles e.g. neutrinos
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Yes...LeSage called them "ultramundane particles"...apparently whose origins lay outside of the Universe, wherever that might be!!!. That puts the kybosh on the theory straight away.
Not only that, but when you think about it, the flow of the ether, the presence of these particles, would cause weird optical aberrations with light traveling through that ether. Not only from the changing nature of the gravitational field from point to point in such an universe, but also from diffraction caused by photons bouncing around off those particles.