IF that were true, Abu Ghraib never would have happened. You remember that, right? Where Americans tortured people? Where they took selfies laughing next to them?
Extending their visa via sponsorship would mean what? What does that require? Also, having an H1B visa doesn't qualify anyone for SNAP benefits. To be an immigrant and qualify you have to be of a specific immigrant group (cuban, haitian, etc) or be a green card holder AND have been here for 5 years with that card.
All of that requires what? Being documented.
There are not huge numbers of illegal immigrants stealing jobs from Americans then somehow qualifying for SNAP benefits.
This is a conservative racist fantasy. Just like the one Reagan pushed back in the 80s with the "wellfare queens" with 11 kids who got Cadillac's paid for by the US government.
It peddles in the same "they're stealing from you" nonsense.
And if you took 15 minutes to research how the benefits system in the US works, you'd know that.