Don't make the mistake of confusing rhetoric with action. Republicans haven't meaningfully shrunk the size of government since Eisenhower. Nixon grew the government substantially (EPA, HHS, Dept of Education), Reagan talked small-government and acted differently, HW stayed the course and W was a "compassionate conservative" who added what was biggest healthcare entitlement in the last twenty years (Medicare Part D).
Most Republicans are in favor of public schools and, in general, of having a much larger government that a night watchman state, including a pretty robust social safety net.
You cannot call these people "anti-government."
Look at George W. Bush. A good example of a Republican. He certainly expanded the US government, which (in my opinion) was already massively oversized.