The government is accountable (in theory). The government is a different kind of entity with different behaviour to a single human, or even a corporation.
Your argument is a semantic one and was not made in good faith I think...
But if the government acted more selfishly , like a corporation (it's heading that way), then yes you are also competing with the government in certain areas. It partly depends if the billionaires in question and your semi-fictional idea of government are colluding directly. In which case you would be competing with a n-trillion dollars of capital for things like housing, some of which is controlled by billionaire beneficiaries. Essentially government monopolies are what you might be worried about, which do exist.
In reality the government also spends some of its money on infrastructure and other common-goods, which creates common wealth. The government (with central banks) also creates money so the idea of direct competition (which makes no sense to me outside of something like sovereignty over large amounts of land / mineral wealth / taxable subjects ) isn't so relevant.
