Sorry, you're wrong on all counts. Conservatism has said it's for small government (except of course when it comes to policing women's bodies or who consenting adults sleep with). Authoritarian governments don't have to be big - it's harder for an autocrat to control a bigger apparatus. Smaller governments are easier to pack with loyalists. Fascism does not require big governments, it only requires a government - in any form - that follows fascistic tenets.

And it's on the far right of any spectrum - the fact that it doesn't seem "so" far to the right here in the US is thanks to decades of Republicans moving the Overton window steadily rightward, to the point where fascism seems OK with a large number of people.