Let's put it this way - if it wasn't for private entrepreneurship and Musk's vision, SpaceX wouldn't be here no matter what DARPA or NASA would have done.

However then they did got significant (200-300M) in funding because of the effort around commercial resupply of ISS and more later but that financed the whole Falcon 9 and Dragon program.

That funding they used to offer a rocket that then accumulated a huge backlog of launches because of the low pricing they offered. Note, this was before re-usability was even a factor.

Its also true of course that other companies, including ULA (and its parents LM and Boeing), Orbitl ATK and others also had to option to commercially bid for those government contracts.

Orbital ATK did however their rocket met with no success in the commercial market.

Rocket Plane Kistler tried, but their efforts were very much Old Space and they couldn't raise the insane amount of money they wanted to raise and NASA kicked them out of the program.

Overall, SpaceX was certainty helped by the need for Commercial resupply of ISS and without that they might have gone bankrupt. However, what they did with the funds, was truly amazing and beyond what anybody could have hopped for.