Yes, NASA may partially rely on SpaceX in the future just as it's been relying on Boeing and thousands other contractors, so whats so new about that?
You actually know what's so new about it I think
SpaceX by far isn't something like Boeing, Lockheed Martin etc. who work for the government anyway (designing and building war machinery in the first hand). NASA will have to rely on something that is "not invented here" - Falcon9 + Dragon. This is somehow historic. All earlier programs depended on hardware which was developed hugely under the pencil of NASA engineers and scientists. But not so Falcon9 and Dragon. It's not designed by NASA and not build by many huge companies but by a small private company which intenions is, and already became alive, to push commercial space flight forward. SpaceX even operates its own mission control. Not so Boeing, nor Lockheed Martin or others. We have a completely new situation here. We have two different bodies which perform space flight - a huge governmental body and a small company. I.e. SpaceX is not just a subcontractor like Boeing or the others.
And why does it matter if NASA was created as a result of competition from the Russians? Almost every major exploration - definitely those mandated by governments - was a result of some form of competition - political, scientific, or both. That doesn't change the mandate and purpose of NASA, which is Space Exploration, Administration and related R&D.
Yes, these days the true mandate of NASA is R&D luckily, and NASA does it in a way and scale like no other space agency. No doubt. But back in the late 1950's and early 1960's R&D was just a kind of smoke screen, just like it was the case in Russia as well of course. The first and foremost intention was arms race/saber-rattling but nothing more. That is what I wanted to say. Wernher von Braun did invent and revolutionize rocket technology, but he sadly did so as a war criminal. Also something people should never forget (or some even don't know). There is a Pomp and Circumstance, but also shadows. And if NASA wouldn't be that much important politically (it keeps a lot of jobs and important votes), it would not exist anymore in the way it exists.
The thing is that in fact something like NASA is not mandatory overall to explore space. It is just one way how it can be done and how it was done in the past. In in this way it's done in some kind of a monopoly, even preventing innovation and something like a market. But NASA also could be handled differently and much more efficiently. Having talked about it to a lot of space flight enthusiasts and even people inside the business, this is what I hear all the time for almost 10 years. And those people were right. They even were right about Constellation and Ares from the beginning.
When Apollo was being cut by Nixon (another reason why space flight should not overall depend on 4 year periods) a lot of people left NASA disappointedly and went into private businesses (both aviation and space flight subcontractors). From these days on a lot of former NASA people and astronauts talk about commercial space flight, with good reasons. Because commercial space flight industries would enable much more sustainable stuff than a big bloated agency, which always depends on clueless minds who change office and seats every few years. Space flight needs to be available to a much wider range of people and companies. Just like it happened to aviation. Do we always want to sit in front of our PC's and watch NASA launching tax money into space? I rather would take a seat as well
Well, and I wouldn't say that no company would fly to Mars within our lifetime. I expect to live for at least another 60 years. And when one looks back 60 years, one can expect amazing things to happen within the remaining lifetime, well depending on ones age of course
Something like SpaceX was unthinkable in the 1960's. Sure, they get money from the gov. as well. But they need less to do almost the same as NASA (launching cargo and men into space soon), and they don't do it with design and logistical help basically from NASA. They are able to contribute to space exploration in future, by less tax money, and even by making money. This is "why" private companies will be able to perform cutting edge manned space exploration.
The current situation in Washington and inside NASA and SpaceX, is a perfect real life example/proof of how things can be done in a different way without spending 50 billion tax dollars for something like Ares that carries 50% less than the Shuttle by being as twice as expensive operational (and this just to reach LEO i.e. the ISS).