It's pretty simple. The theory of general relativity models gravity as local spacetime curvature. With respect to this model gravity IS local spacetime curvature and IS considered to exist for the GR domain of applicability. So a absolute answer that it exists, or doesn't exist, is irrelevant. It exists in GR but it doesn't exist in Newton's Law of Gravitation. Personally I think we directly measured the geodetic effect, spacetime curvature, during the Gravity Probe B experiment. If you want a yes or no answer then ask me whether the predictions derived from EInstein's theoretical model have been empirically confirmed when tested. The answer is yes.

Infinitesimal means very, very, ....... small. So small that when we connect the dots tangent to every point on the spacetime manifold we find out that the local spacetime can be flat over very large areas. Even in the strong field the local spacetime is flat over the segment where you're at. At the very least. The choice of coordinates used in an analysis, what scientists do, is a function of what's appropriate for the goals of the analysis they're working on. It's about modeling natural phenomena not finding absolute answers to questions that can only be confirmed or falsified each time the questions asked.