SpaceX has negotiated some sweet deals to use existing government facilities
Sweet or not, those deals were negotiated in a competitive environment. SpaceX recently inked a 20-year lease on SLC-39A (intended for their upcoming Falcon Heavy), and in addition to the rent, they are paying for all the upgrades and renovations they have in mind. (IIRC, Bezos also wanted pad 39A, and they had a bit of a bidding war over it.)
SpaceX has received a lot of seed money from NASA
Not sure what you mean by "seed money" here. Yes, they got a COTS contract (also in a competitive bid) to develop their hardware, but so did Orbital. How is this different from, say, Grumman getting a contract to build the LEM back in the 60s?
I don't mean to imply that SpaceX did it all without any help. Musk knows full well he's standing on the shoulders of giants, and readily acknowledges that they couldn't have succeeded without NASA's help.
That said, I fully agree that being a private company is one of the main reasons why SpaceX is able to beat everyone else on price. I would also point to vertical integration as another reason. They're not trying to integrate thousands of parts and modules from a supplier in every Congressional district. Except for "commodity" parts, they make most stuff in-house. This alone gives them tremendous agility, compared to any gov't program.