“I don’t see how it would follow that we would all experience the exact same gravitational effects as everywhere else.”
Well, you’re right that the field would lessen with distance, but if every particle in the universe was emitting a gravitational field that had infinite speed, you would see an effect of interference patterns forming as the gravitional fields would be colliding with every other field. The out waves from all the other fields in the universe would be surrounding every other particle as in waves, creating standing wave patterns. Those standing wave patterns would then harmonically resonate with the other standing waves surrounding all the other particles, and eventually, if the universe was old enough for the effect to be fully realized, the gravity would be the same in every location in the universe.
That sounds really cool!
I'm afraid I still don't get how gravity would be the same everywhere, because you'd still have localized gravity overpowering any other gravitational force. IIRC, gravity is the weakest of all the known forces. It's only because the inverse square law doesn't apply that the cumulative effect of gravity overpowers every other force, given enough mass. Sadly, it's been a long time since I've looked at any of this in any kind of detail, so I'm probably way off base here. Thinking about gravity makes my head hurt.