There are a lot of assertions in your post. I'll approach this by addressing them one by one.
Who are these "people" you are refering to? What do they mean by "gravity is not a force"? What do you mean when you use the phrase "natural motion"?
I don't see a problem with that? This is where the idea of a field came into use in physics. Regarding your example, a similar thing happens when dropping a charged body in an electromagnetid field. The way these phenomena are described in modern physics is that as follows:
Source generates field at R = R(x,y,z) -> field acts on body which is located at R = R(x,y,z). Body at R = R(x,y,z) accelerates according to field strength at R = R(x,y,z).
In the case of gravity the source is anything that has matter, i.e. something with non-zero active gravitational mass.
In the case of electrdynamics the source of the electromagnetic field is anything which has charge and/or current.
Sorry but I have to cut this off here. I keep falling asleep at my desk.
It cannot be correctly argued that "gravity is not a force" just because it is a geometric theory.
From Albert Einstein, in a letter to Lincoln Barnett (1948), wrote
The concepts of Physics have always been geometrical concepts and I cannot see why the gik field would be called more geometrical than f.i. the electro-magnetic field or the distance between bodies in Newtonian Mechanics. The notion probably comes from the fact that the mathematical origin of the is the Gaussian-Riemann theory of the metrical continuum which we are wont to look at as part of a geometry.