Georgia HB 87 (2011) and its unintended consequences say otherwise. While the H-2A is indeed popular, there are still a staggering number of people working these jobs who are here illegally.
Indeed, in my country, and despite scaremongering about boats, 95% of illegal immigrants are illegals because of visa overstay, and I'd bet the US numbers are around that.
There are a massive amount of overstays on H2A & H4 visas. They didn’t arrive here illegally, but their current status is very often illegal as they can’t switch jobs or extend beyond a season without a cumbersome renewal process, often which requires leaving the country for at least 3 months.