> Public systems however have the same price pressure than private systems, since people put this pressure up via democracy.

Not at all. Far from it. This is a very naive view of voting and public choice systems. See for example:

https://en.wikipedia.org/wiki/Public_choice#Special_interest...

https://en.wikipedia.org/wiki/The_Logic_of_Collective_Action

https://en.wikipedia.org/wiki/Government_failure#Political_s...

https://en.wikipedia.org/wiki/Regulatory_capture

https://en.wikipedia.org/wiki/Public_choice#%22Expressive_in...

https://en.wikipedia.org/wiki/Rational_ignorance

https://en.wikipedia.org/wiki/Principal%E2%80%93agent_proble...

https://en.wikipedia.org/wiki/Jerry_Pournelle#Pournelle's_ir...