Actually, infinite velocity is exactly what it implies IMO. I don't see how it would follow that we would all experience the exact same gravitational effects as everywhere else. Gravitational effects are still bound by distance. Not inverse square, thankfully, but the effect lessens with distance, though in theory every single particle, no matter how separated by distance does have some effect on each other gravitationally.
I can't remember exactly where I first read about the discrepancy between the sun's apparent position and it's gravitational effect, as it was some time ago. I think it was a freeper who pointed it out to me, but I get old and forgetful. I'll do some searches on this tonight, if I'm not distracted by shiny things.
“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.