Musk doesn’t put a whole lot of importance on degrees. SpaceX doesn’t hire idiots, and people who aren’t idiots are pretty likely to have degrees, but that’s correlation, not causation. Tom Mueller designed the Merlin engine, its variants, and a few other engines along the way. He has a BS and Masters in mechanical engineering, but that’s not why Musk hired him. Mueller was hired because he was building turbopump rocket engines in his garage and firing them out in the desert. His day job was designing novel engines for TRW that never flew, but lots of people in aerospace have that kind of position. To go home and build your own engines because that’s the only way you’ll see your own work fly takes a certain obsessive quality that Musk was looking for.
As an aside, I think “rocket scientist” is a bad name for the position Musk holds. “Rocket engineer” is only slightly better. I would say “rocket architect” is good, however. A normal architect designs a building at a high level, with the low-level details to be worked out by others. And yet, the architect must have a solid grasp of what is possible, given the materials, current state of the art in manufacturing and engineering, etc. The architect must also make economic decisions, and be aware of what cost tradeoffs are possible. A rocket architect is in a similar position, except that the margins for rockets are so much narrower that the architect must have an even keener grasp of the important details, even if they are still not designing any particular part.
Sergei Korolev and Werner von Braun would be historical examples of rocket architects. Musk is probably the equal of either of them at this point. He does have the advantage of having complete control over SpaceX, whereas others had to contend with political infighting.
Everyone who has worked with Musk, even people who have had a falling out or left SpaceX on less than the best of terms, has said that he is able to grasp new subjects very quickly. Robert Zubrin, for instance, said that when he first met Musk, he had essentially zero knowledge of rockets, and it showed. A short time later, they met again and Musk knew everything there was to know on the subject. That doesn’t mean he was designing rockets down to the nuts and bolts, but that ability to absorb vast detail did make him an excellent rocket architect.
The Falcon 9 is in many ways nothing special, but they made good decisions on just about every front. Choice of propellants, ease of manufacture, materials, engine commonality, horizontal integration, even things like diameter (chosen to be the maximum transportable by highway) all made for a very well-balanced rocket. There was no outside influence forcing them to use solid boosters or Russian engines or to put their factory in Mississippi. Nor did they shoot beyond what they could reasonably achieve (no spaceplane, no single-stage-to-orbit, etc.). Musk and his crew just made a nice clean-sheet rocket that didn’t do anything stupidly, and amenable to a great deal of constant refinement.