Not fully, you need to add that the legacy airline is funding Jet Blue. Elon Musk explicitly said SpaceX would have not been able to achieve its goals so fast and so cheaply without the access to all the data from NASA. Basically, SpaceX was able to "ride" on the prior work of NASA, while having funding from the NASA. It is more a bit of a mixed spin-off/independent company in this case.
Nobody is saying they shouldn't have access to it. They're saying that touting Space X as an example of why government is bad and private industry is great is misleading.
What Space X is a good example of is the very effective model we have here in the US of doing innovation. Government funds fundamental R&D, then private industry commercializes it. It's a healthy synergy that you see throughout the sector.
I think that the more important question is, when another company wants to do space exploration in the future, will they also get access to NASA's expertise? Otherwise, the taxpayer's money has been used to create a barrier to entry for the (future) competition.
Most of the information is technically available to the public. Some is already online, some is available for the asking. Some is probably classified (I'd be surprised if SpaceX was given much if any of this), and a lot of it almost certainly is under legal restrictions regarding transfer to non-US entities, but that is a limited barrier to an American company that, of necessity, would have hundreds of millions of dollars in funding.
