If I understand correctly, to prove that problem $A$ is NP hard, you need to pick all possible problems $B_{i}$ that are in NP and then prove that they reduce to $A$ by using a polynomial time computable function, that maps instances of each $B_{i}$ to instances of $A$.
Once you find the first NP hard problem, by using reductions you can then find that many other problems are either NP Complete or NP Hard. However I imagine that this depends. If you are unlucky, then maybe all $B_{i}$ problems reduce to $A$, but $A$ reduces nowhere else, so your proof is essentially useless.
My question is about the motivation that Stephen Cook behind showing that the SAT problem is NP hard. Did he see a lot of potential behind this problem? Did he know that if he showed that this problem is NP hard then a lot of other problems could be shown to be NP hard as well?
In short, what is the story behind this proof? Because after studying some basic complexity theory, it really seems like this proof was one of the most significant ones in this area.