Sure public money is tangentially involved in everything sciencey, because private industry has a terrible time financing science. But one reason the public finances science, is so industry can grow and thrive and remain competitive. Imagine the other approach: the government 'owns' every innovation and controls industry. Not a pretty situation.
Btw Bell Labs, not strictly funded by the public. And no phone runs on Unix today. Never mind 'linux' is spelled sort of the same, they are different animals.