All forces are points.In particular, gravity is a point source.If you wish to distinguish between falling and weightlessness suspend two balls in front of you.If they move, ever so slightly, toward each other then you are falling.If they do not, then place them in the opposite direction and repeat the experiment. If you visualize a sphere, by positioning the balls diametrically opposite each other around the sphere you should be able to determine the direction of the gravitational force, or at least the maximum one.The balls are following their V path to the center of gravity. It should be very unusual to be actually weightless. If they are so far out that the movement can be explained by their mutual attraction then changing the direction should produce exactly the same movement.
In order to do any math we make use of gauge invariance.That is we say that for all practical purposes we can ignore this curvature.For all practical purposes, until very recently, the earth is flat.By itself, this is perfectly fine.But then we turn around and apply these derivations to the universe.
I propose two experiments.They both involve sending a probe as far from the sun as we can communicate with.Once there repeat the measurements for the red shift and the microwave background radiation.I believe that our proximity to the sun is a factor in these measurements.