Here is how I see it. The equivalence principle could also said to be foundational to Newtonian classical gravity, and it is only associated with general relativity for historical reasons.
General covariance is a mathematical principle, and it is not clear that any mathematical principle, by itself, has any physical consequences. As Einstein is quoted:
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.The laws of math do have consequences, but they require interpretation.
We normally say that Euclidean geometry has physical consequences, but that is only because of some assumptions about how idealized points, lines, planes, etc. correspond to physical entities.
Historically, covariance under Lorentz transformation was crucial to the presentations of relativity given by Poincare in 1905 and Minkowski in 1907. Einstein had a weaker version that was more similar to what Lorentz published in 1895.
Einstein was unconvinced, and wrote a paper denouncing the concept in 1913. Eventually, Grossmann, Levi-Civita, Ricci, and Hilbert convinced him to accept it.
The main ideas behind relativistic covariance had already been published by Poincare and Minkowski, but they were dead by the time Einstein worked on general relativity
I believe that general covariance is a crucially important concept. It is a way of saying that the laws do not depend on coordinates. That different frames can be used to the same effect. These ideas go right to the core of why the theory is called relativity. And historically, covariance was a large part of why relativity was accepted. By 1908 it was widely accepted, even though Einstein did not.
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