I could give you an entire thesis on Hitler's 'privatization' and the NSDAP economy post-MEFO bills - Schacht and MEFO bills were the only existing element of state-enforced privatization prior to the 40's or so, but to state that no independent privatization occurred at all is just fucking hilarious considering the German economy (while being compliant with the state, a concept that makes sense..a nation's economy should be wholly subservient to the nation that hosts said economy and the interests of said nation, unless you prefer deregulated economies) was essentially ran by things like I.G. Farben, Krupp and FW - Farben, despite being "forced" to parrot the party line, had little issues with going the way of the German state at the time, along with many other German industrialists; the mistake is believing the large majority of the German private sector at the time didn't want what the NSDAP introduced, because it did.
As for the rest of what you said, yes, fascism is a totalitarian, syncretic system that meshes elements of the left & right - D'Annunzio and Gentile both coined the phrase 'far-centre' when considering fascism and it's placement on the dualistic spectrum, and Gentile was the actual author of the Doctrine of Fascism; Mussolini's contribution compared to Gentile was fairly mute, and he was mostly an attendant ghostwriter for the book; fascism's earliest influences came out of syndicalist economic movements and drew from people like Georges Valois - the 'association' with socialism mostly goes that far: former fascists came out of syndicalist economic movements, thoroughly disillusioned with the internationalist and socially left-wing elements of far-left ideology; the entire shtick of fascism as an ideology and set of principles was to recognize the importance of an economy based on the working class and ensuring the working class was properly compensated for their work and had a proper place within the state without also pitting the working class against the rich and recognizing the sheer fucking idiocy of 'class warfare' notions that communists create - most fascists propped up class cooperation, as opposed to class warfare, and this derives from the Spengler-influenced perception within fascism of society as a living organism that has to be treated organically.
'Fashcap' exists, fascist capitalists, but they're as fucking funny to me as ancaps - fascism is inherently built to be opposed to capitalism and capitalist economic theory, but only to a point and that 'point' exists to prevent capitalism from reaching the grotesque, unregulated hyperactive stage it exists in today; in other elements, there is no problem in fascist doctrine with the use of capitalist economic elements in the creation of a foundation or the usage of the private sector - the complete removal of the private sector is retarded, unsustainable, toxic, and unrealistic; if you're attempting to make the argument that these fascists aimed to create a state economically similar to the 'all-equity, raceless, stateless, utopian all-worker commune' communists envision, I will debate you endlessly, friendo.