Here is the situation in Europe :
Ariane 5 launchers are being made by Airbus Safran Launchers.
Airbus Safran Launchers is owned 50%/50% by Airbus group and Safran group
Airbus group is owned at 26% by government
Safran group is owned at 15.4% by government
So Airbus Safran Launchers is owned at 20% by government, this is far from being a government entity.
Arianespace launches Ariane 5 (and others).
Arianespace is owned (well, will be soon) at 75% by Airbus Safran Launchers and at 25% by other private industrial entities
Therefore Arianepace is owned at 15% by government, this is far from being a government entity.
The fact that government buy rockets to send in space some military satellite is true. But saying that because governments use to buy rocket make the entire rocket industry a government entity is like saying that because government buy trains, the entire train industry is a government entity.
Of course government put a lot of money into the space program, but this is to ensure the autonomous access to space! But you know that (at least in Europe), the space industry makes the majority of its money thanks to commercial satellite?
Also, so that you could freak out! Do you know that Airbus Safran Launchers is responsible for the M51 manufacturing? Because yes, ICBM are the same technology as rocket. So why not use the people capable of manufacturing it to do so, and yes, this is a private industry!