The reason its generally classified as a conservative ideology is mainly its focus on social conservatism. Ideas like economic heirarchy, general traditionalism, and religiousity while not uniform across fascist societies were generally really common. Also nationalism and imperialism tend to be ideas associated with conservatives. But your issue seems to be moreso with the economic side of things. While it is true that fascists were interventionist in some regards like establishing various social programs, they also privatized a massive amount of their economy. Corporatism was a huge part of fascist economies with corporations giving large control of parts of the economy.

Also its probably worth noting that ideas like liberalism and conservatism have changed significantly over time. A 19th century conservative probably would be a monarchist and semi-feudalist, a far cry from modern conservatives who lean more towards what a 19th century liberal would be.