1) SpaceX employs engineers who worked on the Shuttle, thus getting all of their lessons learned for free.
2) SpaceX gets to use facilities NASA built, like launchpads and testing facilities much more cheaply than if they had to build their own facilities. They tested their heat shields at NASA Ames, for example.
3) Everything is cheaper now. Not just raw materials, but imagine the cost savings when every engineer has a (Shuttle Era) supercomputer on their desk. Testing is dramatically less expensive.
4) Everything is better now. The (NASA developed) heat shields are much higher performance than those available in the Shuttle era. The level of computer control is much more sophisticated. Better technology means you can do more with less.
In short, SpaceX has massive, massive advantages over the original Shuttle team. I would hope that they could do more, more safely, at less cost.