Using modern terminology, Hamilton was an uber-mercantilist or "neo-mercantilist" as he and Friedrich List are often called. He was not a capitalist in the modern sense of the term, which is synonymous with an anti-interventionist laissez-faire attitude (although some 18th century users of the term "capitalist" employed it to refer to an interventionist who uses the power of the state to redirect wealth to himself or his allies, in which case it probably would apply to Hamilton).
There is no capitalism which exists in the theoretic world with on intervention. Never was and never will be for many reasons. Ayn Rand is not an economic thinker of high quality or historian.
Hamilton's ideas went far beyond mercantilism which was based upon the concept of robbing Peter to pay Paul. He understood that the economic world was not a zero sum game and that limiting an economy to the gold supply would not work in America. A Central bank or National bank was anathama to mercantilist theory.