I agree. Moreover, socialism and capitalism are not antithetical--they are orthogonal. Norway and the US have socialism (e.g., Social Security) and capitalism (e.g., people invest their capital to fund industry).

Socialism without capitalism is communism--the state owns everything.

Capitalism without socialism is anarchy: if you don't socialize a legal system, law enforcement, and national defense, then you don't have a country.

East India company and its armies are something of a counter argument to your second point.

Anarchism exists within socialism too-- the anti Francoist anarchists in Spain who were crushed by Soviets for doing the wrong kind of socialism, for example. Kropotkin's approach and anarcho syndacalism is anti-statist, and also anti-capitalist.

Anarchism is a complex area that can be capitalist, socialist, individualist, collectivist, primitivist, techno..logist? Basically, like communism, anarchism doesn't have just one flavor.