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.
Well, the way I’m conceiving of it, it would be something like the tidal effect. If you had two moons, we might have more than two tides per day, but we’d still have the same number of tides all over the planet, because the tides would be due to the net effect of both moons together.