Alternate definition of riemannian manifold

Question

By the Nash embedding theorem, it seems the definition of riemannian smooth manifold is equivalent to the zero locus of some functions $f_1,\dots,f_r : \mathbb{R}^n\to\mathbb{R} $ which are $C^\infty$ and which differentials are zero at no point.

This alternate definition is simpler, because it does not introduce partial charts, the metric tensor is just the restriction of the euclidean inner product and the Levi-Civita connection is just the orthogonal projection of the usual derivative of vector fields. And as a bonus, the zero locus approach is closer to the definition of affine schemes in algebraic geometry. Pseudo-riemannian manifolds can also be defined this way, under the additional assumption of stable causality.

So what is the advantage of the standard definition of riemannian manifold through charts and custom metric tensors? Is it just that the dimension increase of the Nash embedding $n(n+1)(3n+11)/2$ makes computations intractable? The story of an internal observer's viewpoint is not very convincing, because the only riemannian manifolds with rigid body motions (as would be a true internal observer) are the ones with constant curvature, so a tiny part of riemannian manifolds.

Answer 1

The comments note multiple obstacles to basing Riemannian geometry on manifolds embedded in Euclidean space. In the hope of distilling major objections for posterity:

0. Intrinsic geometry is a thing. (Before Gauss's Theorema Egregium this was not apparent. The fact tends to get hidden because nowadays we teach Riemannian geometry starting from intrinsic geometry, e.g., Kobayashi-Nomizu Volume I.)
1. Defining Riemannian manifolds in terms of embeddings casts intrinsic geometry in terms of extrinsic geometry.
2. In a particularly strong sense, intrinsic geometry of a Riemannian manifold (traditional Riemannian geometry) does not uniquely determine extrinsic geometry (isometrically embedded submanifolds): Every Riemannian manifold has an infinite-dimensional space of isometric embeddings in a sufficiently high-dimensional Euclidean space. (Twice the Nash dimension is enough to bend the ambient space into an arbitrary Cartesian product of plane curves.)
3. As multiple comments note, extrinsic geometry is Difficult To Work With for specific, interesting Riemannian manifolds, does not play nicely with Riemannian quotients and submersions, and flatly does not accommodate other geometric structures.

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To reiterate my comment: The Nash embedding gives existence but not uniqueness. Locally, many surfaces (such as those of constant Gaussian curvature) in three-space are flexible, with an infinite-dimensional space of isometric deformations. For one thing, that throws a wrench into "the Levi-Civita connection is just the orthogonal projection of the usual derivative of vector fields": This proposed definition depends on an embedding, which is Far From Unique.