Einstien's principle of equivalence

Einsteins principle of equivalence, as usually explained, seems strange (to
me) from the point of view of (pseudo) Riemannian geometry.
As explained in some relativity texts, the principle states roughly that
there is no (local) physical difference between effects due to an
accelerating frame and effects due to a gravitational field. However,
choosing an accelerating frame seems like just a certain choice of
curvilinear spacetime coordinates. But isn't gravity supposed to be
curvature due to a metric tensor? But whether a Lorentz manifold has zero
curvature or not certainly doesn't depend on a choice of coordinates. How
can I produce curvature on a flat spacetime just by a choice of chart? I
can't! The whole thing seems especially weird since metric and curvature are
defined as geometric objects with an existence that doesn't even need a
chart at all to make sense.
So I don't see what the equivalence principle could really amount to.
What am I missing?

The principle of general covariance is also a problem for me. Does it amount
to anything more than the statement that the laws of gravitational physics
can be expressed in terms of tensor fields? It seems to me that, all sorts
of ad hoc laws could be expressed in this way. In fact, if spacetime had
trivial topology then one chart would do and any law written in that chart
could just be expressed in any other chart by brute force anplication of
tensor trnasformation laws. In this case the idea of general covariance
starts to seem empty.
What am I missing?

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