> What's the physical content here?

The physical content is that measurements of distance and time do not depend on the choice of coordinates. 

> even if we assume that the four indices of 𝑑𝑥𝜇 in GR refer to actual spacetime-measurements by an observer, 

No, that is not a valid assumption. The $dx^\mu$ are not measurements, they are just labels. The actual spacetime-measurements are $g_{\mu\nu}dx^\mu dx^\nu$