At this point the whole "small government" thing seems to just be a dog whistle for the Republican base that roughly translates to "if it helps people I don't like we won't fund it."
At this point? Nah, in my view it started with Reagan's "welfare queens" (demonize the recipients of "large government"), which is going on 40 years ago. And that's just because it happened within my lifetime. One can probably go back further, perhaps to Earl Butz from the Nixon administration.