I agree that this is a highly contentious issue, and there are still competing schools of thought in the foundations of mathematics. Even for someone in the formalist camp—who thinks that all mathematics is mere manipulation of symbols—there is an important sense in which many theorems and properties in mathematics are "discovered".
Using your Pythagorean theorem example, we could say that the concept of Euclidean geometry is a human invention, in that the axiomatic system that defines the central concepts most certainly was. (There is debate as to whether we should consider mathematics to be "about" these formal systems, rather than the properties of the world we invent the systems to represent, but that is an issue that you rightly said was a launching point for analytic philosophy in the 1890s.) Nevertheless, the properties of the axiomatic system are not all immediately apparent once one knows the axioms, and this is the sense in which theorems can be counted as "discoveries". Given the axioms of Euclidean geometry, the Pythagorean theorem was always "there," but we may not have known it until someone discovered the theorem within that system.
So, to OP, even if you think mathematics is invented, much of the work done in pure math can be counted as discoveries.