To get rid of the point in the center, let's try another approach. We cut the Earth in two, right across the equator, and separate the two halves by a short distance. And we get rid of all the air. Can't have anything interfering with our thought experiment. Now our orbital velocity will increase for a while as we get closer to the center, but then it will start dropping off because there will be more mass above us helping to hold us up. When we got to the center, our velocity would be zero.
Of course, we couldn't do that because we have no method of supporting a zillion tons force. So then I got to wondering how big a sphere could we actually build? Could we build one large enough that it's combined mass would create a gravitational field strong enough to hold it together? Hmmm.
Any amount of mass is going to have some effect on the gravitational field, so two rocks with polished surfaces placed next to each other in space are going to exert a microscopic amount of pull on each other, so barring any outside influence they should stay stuck together.
So I guess what I'm looking for here is, how massive could we build something with a hollow center? We aren't going to be able to hollow out the Earth, the center is molten iron. We are able to dig mines that go a mile or two into the Earth, but we need to blow cold air into them to keep them habitable, but 100 miles down the temperature is much hotter and the pressure is beyond our capabilities.
If we stick to one gravity, because that's how much force a Roman Arch can withstand, and because we would need a certain amount of force to hold this thing together, then maybe the question is how big of a hollow sphere could you make using the same volume of rock / iron that is found in our planet? Now the question becomes how thick do you want to make the skin? If the skin is a thousand miles thick, then the outer diameter of our sphere would only be about 7,000 miles, or about twice what is now. But a thousand mile thick skin is still going to have problems with heat. I suspect compression is going to generate enough heat to make it molten.
If we decrease the skin thickness to one mile, our outside diameter will roughly the same as the moon's orbit. At that distance, the force of gravity will have fallen off a bit, but not that much, so you could count on that force holding everything together. If you have that much gravity, then you should be able to have an atmosphere, but covering that large of a sphere would take an enormous quantity of air.