If the answer you're looking for is, "all money ultimately originates with the private sector" then that's completely wrong, isn't it...
How many private sector companies have large contracts with various govt bodies? The likes of Dell, HP, Capita, etc, etc.
The % of public sector workers in this country is just under 20%. That's a lot of us. The private sector will be receiving a decent chunk of cash both from govt and from public sector workers spending their own pay packets.
Money is a vast and tangled web*. The private sector did not invent money nor generate the money in circulation, did it. Not quite sure what your point is.
*More like a circulatory system.