Who is removing people's rights, and how? Although government exists, the private sector has a strong public welfare directive, e.g; religious charity, science research for improved quality of life, creating long-term investment and partnerships, and a diverse market for people with diverse illness.
To govern is synonymous with to manage. What difference are you trying ascribe? Is there some service the private sector is incapable of providing altogether?
Why must taxes be reduced? This doesn't consider the cost of the government now having to regulate the new contracted services, nor the cost of doing business with them, nor the possibility that industry will adapt to the economy and attempt to increase their margin knowing the cost isn't their own.
A private company could attempt to reduce prices to essentially nothing (and effectively much lower than tax) by fulfilling a niche bundle in the market such as labor organization, patient studies, sustainability & investment, self-defense, journalism & investigation, etc.