No, because he wouldn't really be running NASA the way he runs SpaceX. Government organizations are subject to a host of political influences that that don't affect the private sector. I used to work for the Library of Congress where the Librarian of Congress at the time was a political appointee who spent less time managing the collections and more time updating the carpets.
NASA has worse problems because members of Congress see is as a tool for funneling pork spending. Any director of NASA would have to spend time playing politics with Congress in order to keep his funding and that would undermine any coherent vision that he brought to the enterprise. If buying rocket engines from Louisiana would save NASA billions in the long run, but funding depends on buying them from Alabama, Alabama it is. High-level government bureaucrats must be career fundraisers first and visionaries/competent managers second.
As the sole owner of his own launch company, Musk as a degree of freedom that he could never get as a member of the government bureaucracy and it's that freedom that allows him to run SpaceX as well as he does. In lesser hands that freedom would be a noose, but at least in that case the cost of failure to the public would be very little.
>Government organizations are subject to a host of political influences
The private sector is not magically free of political influence, and SpaceX feels it stronger than most due to its reliance upon what is effectively a government monopsony.
I think if he were running NASA it would probably be run pretty well.