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.
But doesn't it make sense that less government = more freedom and more incentive to achieve?