No.
"Small government" has absolutely nothing to do with the conservative movement, or why people have voted Republican for the past 50 years.
I mean you're right that Republicans would not have won without racism, religious wedge issues, and militaristic fearmongering, but there ARE millions of people, many of which that I know personally and in fact I actually used to be, that believe/are convinced that government is inherently inefficient and should therefore be small.
That number is not inconsequential and is the result of remarkable evangelization by intellectuals like Friedman, Reagan, etc.