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Why not? (to the first sentence, locally is locally)

But I think I have found the source of my confusion in my original post, I was thinking about manifolds only in the topological sense and for some reason forgot that when there is a metric then we have to preserve distances. No 'mug and a donut' type of shenanigans.

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As distance to center of black hole approaches zero, curvature approaches infinity. There is no "locally" one can constrain small enough to observe no curvature upon approach to a singularity... Given two particles arbitrarily close together at arbitrary velocity (< c) relative to the center of a black hole, there is a distance from the center at which they will encounter unequal acceleration.

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