Actually, all things considered, we're still pretty much "the little guy" of aerospace (even if social media trends would convince one otherwise).
It is fun to imagine SpaceX as some megalith of the private aerospace industry, poised to usurp NASA and colonise the galaxy in the name of the free market (which is almost the narrative you'd get reading all the press), but this kind of thinking has led to some really unfortunate misconceptions in the past (such as that of Gene Cernan, who did later apologise to Elon once he learned more about the company). The reality is that in some ways we're still basically a mid-level startup (as startuppy as a defence contractor can be, in any case), and without the successes of COTS 1 (December 2010) and COTS 2 (May 2012), most people here would never have heard of SpaceX.
We're definitely not in competition with or on track to replace NASA (they're our biggest customer!), and we're not even the most prominent name in government space launch at present (media/popularity aside). ULA, in fact, is the current "evil empire" of the public launch market (United Launch Alliance; Lockheed/Boeing joint venture founded in 2006). Aside from ULA, there are a bunch of small companies with similar visions (e.g. Orbital), as well as some pretty fierce competition both in the US and abroad gearing up to seriously attack commercial launch / space exploration (e.g. China). Keep in mind that this isn't necessarily solely related to pressure from SpaceX, but also the natural result of a recovery in the global launch market as a whole since the crash in the 90's.
Don't get me wrong: SpaceX is 100% on track to kick everyone else's ass and get us to Mars next decade. However, it's pretty far from a given, and there is still a ton of resistance against new entrants (entrenched deals and interests, competing lobbyists, FUD/propaganda floating around Washington, etc.).
Right now is way too early for you to start worrying about what could potentially happen in a hypothetical SpaceX-dominated launch / space travel market (especially if it's for the sake of being contrarian). Let's at least approach that point in time before seriously discussing it's potential merits and drawbacks.
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Edit / Full Disclosure: I work here, but I don't speak on behalf of SpaceX or anything of the sort.