Gwynne Shotwell deserves a lot of credit for leading SpaceX in its day-to-day operations. Without her I doubt it would be working as well. Everyone else on these teams also deserves a great deal of credit. They are teams, of course.
On SpaceX being government funded -- yes, that's true to an extent, though they have made a lot of money off private satellite launches too. The fact is that the entire aerospace industry has been deeply embedded with the military and other government agencies since its inception. It's a very heavily subsidized and regulated industry.
But team building and capital raising is a skill, and Musk seems good at it -- at least at the beginning. He's great at getting things off the ground (pun intended?) and breaking through barriers of technological learned helplessness. I do get the sense though that he's less of an effective leader as things get more established. Tesla's problems are too boring for him at this point.