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 $g_{\mu\nu}dx^\mu dx^\nu$

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 coordinates are not measurements, they are just labels. The actual spacetime-measurements are invariants like $g_{\mu\nu}dx^\mu dx^\nu$