Yes...but nearly everything ever produced that is run and paid for by the government uses components produced in private industry.
The difference is that if SpaceX fails at something, it impacts their bottom line, where if the government fails at something, it just sticks its hand back into the taxpayer’s pocket!
I don’t disagree with what you said. My point is why is it surprising when private industry succeeds at launching rockets, when private industry has been doing it from the beginning?
As you and I have pointed out, the big differences are who pays for it and who manages it. Also, remember, SpaceX has contracts with NASA, and has had money from NASA to develop their launch capability.