I think you lack a basic understanding of business. If a private company outsources a job to another private company that job is still in the private sector...
You also seem to misunderstand that role of government. It isn't possible for the private sector to effectively and cheaply educate, provide healthcare, infrastructure etc to anyone but the rich. That's why the public sector does it.
So you think the government is providing education cheaply? California spends 52~55% of it's budget on education and it's failing badly. Meanwhile, in the private sector, Apple has iTuneU, Khan Academy, and the University of the People. All free or dirt cheap. A study (I think it was Stanford) showed that the online students outperformed the in person students.
Suggesting that the government is better equipped to provide a product or service than the private sector is simply not supported by the data. The total debt of the US government is estimated to be well over $100 trillion (public debt and unfunded mandates), they are past bankrupt.
I think pubic vs private was proven in post WWII Berlin. People died trying to get away from too much government control. The public school system is union controlled and agenda driven. Charter schools are doing much better.
Compare public sector housing solutions, how well are they doing?
I'm not against public funds for infrastructure, I'm against the waste of the government. BTW, on of the best infrastructure builders is a private company (CC Myers). One reason is they compete, the government doesn't have to compete, they get paid no matter how bad they do.
Look at the management of cities: Stockton, CA. Detroit, San Bernardino, NYC in the 70's because of over spending.
How about the city of Chicago? How are they doing? Per capita, they owe more than the national debt. They have a Junk Bond rating.
How's Puerto Rico doing?
Look at the under funded public employees retirement accounts.
How well did the public sector regulate the banks? 26 different agencies missed the bubble that turned into a global financial meltdown.
How well did the pubic sector do with Bernie Madoff, EnRon, Savings and loans, or WorldCom?
How well did the pubic sector do with Solyndra and many other examples?
The government has failed to educate the people, failed to regulate companies and have run the nation bankrupt, spend every penny of the Social Security money, and are causing businesses to leave the US.
BTW, I have a BS in business and a few decades working as a professional business analyst.