I do think you're asking a reasonable question, but many do not like your way of asking it. It would be better received if you could express it more rigorously and mathematically and showed that you have thought about it more deeply than your wording indicates. After all, this is a research math forum. But let me make some comments.

The first thing is Newton versus Descartes. I have never read Newton's works, so I could be wrong. But since Descartes preceded Newton, I believe Newton must have embraced Cartesian coordinates and used them in his work on planetary motion and the shape of the earth. Is that not so?

As for developing differential geometry without coordinates, many mathematicians, including me, have tried. I'm not sure whether you're talking about surfaces in Euclidean space or abstract spaces known as manifolds. In either case, my impression is that the hardest steps are right at the beginning. First, you need to develop multivariable calculus without coordinates. This can be done but is it worth the pain? Not as far as I can tell, but you can see if you can do it. I definitely could be wrong about that. Second, it's defining what a surface or manifold is.

Some very abstract-minded mathematicians did manage to do this for manifolds, but you lose all geometric intuition and end up in a very algebraic world. Is it worth the pain? Also, not as far as I can tell. After you've defined a manifold, then you can work out the fundamentals of Riemannian geometry using only abstract vector fields. This is demonstrated both in Milnor's monograph Morse Theory and the book by Cheeger and Ebin, Comparison Theorems in Riemannnian Geometry.

As for a surface in Euclidean space, you could first define Euclidean space as an abstract vector space with an inner product. Then you could define a surface to be the level set of a function whose gradient is nonzero and work with derivatives of the function (without using coordinates). The geometry of the surface can now be derived from studying curves in the surface and their derivatives. Some of this is very nice, but some aspects are still easier to calculate and understand using coordinates. In particular, it's difficult to work out examples without using coordinates.

However, in the long run, what professional differential geometers discover is the following: Our main goal is to prove interesting new theorems as efficiently as possible. The most efficient approach depends on the specific circumstances. So we dump the ideology and pragmatically learn how to use all of them. We switch between them as needed. So the fact is that using coordinates is often the easiest way. The basic reason for that is partial derivatives commute. This fact is fundamental and used all the time. Without using coordinates or differential forms (as when using orthonormal frames), that fact is hard to use efficiently.

I do continue to think about it in the context of teaching differential geometry. I do agree that coordinates can often obscure what's really going on. I don't like most textbooks on elementary differential geometry. So I do try to think of coordinate-free approaches that better elucidate the geometry. Sometimes I succeed. Otherwise, it's coordinates or orthonormal frames. Whatever works best.

I do think you're asking a reasonable question, but many do not like your way of asking it. It would be better received if you could express it more rigorously and mathematically and showed that you have thought about it more deeply than your wording indicates. After all, this is a research math forum. But let me make some comments.

The first thing is Newton versus Descartes. I have never read Newton's works, so I could be wrong. But since Descartes preceded Newton, I believe Newton must have embraced Cartesian coordinates and used them in his work on planetary motion and the shape of the earth. Is that not so?

As for developing differential geometry without coordinates, many mathematicians, including me, have tried. I'm not sure whether you're talking about surfaces in Euclidean space or abstract spaces known as manifolds. In either case, my impression is that the hardest steps are right at the beginning. First, you need to develop multivariable calculus without coordinates. This can be done but is it worth the pain? Not as far as I can tell, but you can see if you can do it. I definitely could be wrong about that. Second, it's defining what a surface or manifold is.

Some very abstract-minded mathematicians did manage to do this for manifolds, but you lose all geometric intuition and end up in a very algebraic world. Is it worth the pain? Also, not as far as I can tell. After you've defined a manifold, then you can work out the fundamentals of Riemannian geometry using only abstract vector fields. This is demonstrated both in Milnor's monograph Morse Theory and the book by Cheeger and Ebin, Comparison Theorems in Riemannnian Geometry.

As for a surface in Euclidean space, you could first define Euclidean space as an abstract vector space with an inner product. Then you could define a surface to be the level set of a function whose gradient is nonzero and work with derivatives of the function (without using coordinates). The geometry of the surface can now be derived from studying curves in the surface and their derivatives. Some of this is very nice, but some aspects are still easier to calculate and understand using coordinates. In particular, it's difficult to work out examples without using coordinates.

However, in the long run, what professional differential geometers discover is the following: Our main goal is to prove interesting new theorems as efficiently as possible. The most efficient approach depends on the specific circumstances. So we dump the ideology and pragmatically learn how to use all of them. We switch between them as needed. So the fact is that using coordinates is often the easiest way. The basic reason for that is partial derivatives commute. This fact is fundamental and used all the time. Without using coordinates or differential forms (as when using orthonormal frames), that fact is hard to use efficiently.

I do continue to think about all this in the context of teaching differential geometry. I do agree that coordinates can often obscure what's really going on. I don't like most textbooks on elementary differential geometry. So I do try to think of coordinate-free approaches that better elucidate the geometry. Sometimes I succeed. Otherwise, it's coordinates or orthonormal frames. Whatever works best.