If we had a parliamentary system, which encourages multiple parties and even smaller, fringe parties can have an impact, I would agree with you. But, we don’t.
We have a two party system.
The GOP and Democrat party may technically be private organizations, but the people that control these parties also control who may (and may not) run for political office. In practical terms, they control government, because they control who may be a government official.
This is, by definition, an oligarchy. It is unacceptable for our country to be run as an oligarchy.
Those private political parties receive an awful lot of taxpayer money to still be called private. Your definition is spot on, we are an Oligarchy and in times past they were at least loyal Americans, now not so much.