There are vastly fewer "immigrants for handouts" than right wing media would like you to believe. Coming to the US is incredibly challenging. People who do it are mostly young and wish to work, to support families. Handouts don't accomplish that.
It take tremendous effort to immigrate, legally or illegally. Anyone telling you that they are lazy is obviously lying.
As a US native, I have met zero lazy immigrants, but lazy Americans are everywhere I look. Thus I think this sentiment is more a projection of their own behavior: “they must be as lazy as we are”.
I think you hit the nail on the head. It maps directly to much of their coalition’s rhetoric, accusations, policy agenda, and behavior these days, including, but not limited to, their obsession with pedophilia.