quantum mechanics - How did the operators come about?

This relates a little bit to my previous question (Experimentally, what categorizes a measurement as corresponding to a certain observable?), but it's different in a way and more historical.

One of the postulates of quantum mechanics is:

To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics (Sherrill).

However, absent in the list of postulates is what these operators are. So is the mathematical expression for each individual operator also a postulate that's not listed, or are they derivable from other axioms? Specifically, where did these expressions come from?

Maybe the question can better be answered historically. If I had to guess, I'd say it began with the wavefunction. Some scientists thought perhaps the squared modulus of the wavefunction corresponded to charge distribution or something, but Born was the one who got it right. He essentially made a guess that the squared modulus corresponded to the probability of finding a particle at a certain location. What's funny is that he initially wrote down the wrong expression in his paper (Born), and his paper even got rejected by the first journal he submitted it to, but the right expression is in a corrective footnote. But again, Born simply guessed what the squared modulus meant. There was no procedure; he just thought of an idea and then experiment verified it to be correct.

Thus:

$$\langle\psi|\hat{X}|\psi\rangle = \langle\psi|x\rangle\langle x|\hat{X}|x\rangle\langle x|\psi\rangle = \int d\vec{x}\ \psi^*(\vec{x})\ \psi(\vec{x})$$

So basically in the position basis, the position operator is simply $\hat{X} = x$, which seems pretty intuitive.

I believe the momentum operator came about next; and it was "derived" by fiddling around with the de Broglie relation and just substituting classical expressions for momentum into the Schrodinger equation (can't remember where I read that). The fact that plugging these classical ideas into a quantum equation just happens to match experiment for momentum measurements seems like a pure coincident to me (it's along the same lines as trying to make the Schrodinger equation relativistic; one way of shoehorning relativity into it gives you the Dirac equation and a different way gives you the Klein-Gordon equation).

I imagine the spin operator came about by simple observation. An electron goes up or electron goes down in a magnetic field. Not hard to guess what the expression for that operator might be.

So anyway, am I on the right track here? Have all of the operators come about simply by guessing? And if I'm wrong, how did they actually come about? How would I come up with a new operator for a new type of measurement?