You can see your doctor pretty much freely within the US for nearly any reason. This idea you cannot see your doctor in the US, or that you receive treatment from nurses or administrators is simply not true.
> insurances and especially hospitals
You may be confused, because some doctors operate private practices, while others choose to work for large hospital organizations. Private practices may be more akin to what some people's idea of a doctor should be. Most insurance plans offer a selection of hospitals and doctor networks, allowing you to chose the style and type that works best for you.
In all cases, you can see your doctor if needed, with very little lead time if any.