As we've covered, repeatedly, reduced supply of labor will increase the wages of those poor you're worried about buying food. Based on the first google results I could find farm labor constitutes 10-20% of the cost of food. Even at the high end, if you DOUBLE the wages of farm labor the price goes up 20% but wages go up 100%. And this same principle applies to every sector where new farm labor is pulled from. Wages go up more than prices, so it's a net win. This is great for the poor worker, but bad for the wealthy worker who doesn't get raises due to this increased competition on the low end of wages. To the extent that the wage gap might be a concern of yours, this is one significant way to reduce the wage gap using market forces.

> the big companies will either start selling cheaper (less healthy) substitutes

Millions of new potential customers have extra money in their pocket to be able to afford luxury food products. Meanwhile there are far fewer impoverished people looking to buy food (migrant/illegal workers). So if anything, it increases the market for luxury goods and decreases the market for cheaper unhealthy substitutes.

>import it from places with cheaper labor.

This is only a problem if you believe 100% of the illegal and temporary work visa produced goods could be replaced with imports. Not only is that not true, but even if it WERE true, that doesn't say anything about other sectors predominately filled with migrant/illegal workers such as hotel maid services or construction. My house, in my neighborhood, is not going to be built by someone who isn't physically here.

I feel we've sufficiently covered the initial premise, which is that the policy I've described helps poor middle Americans. You might be able to debate the extent to which it would help them, or that this is good for the over-all economy/country, but it's undeniable that artificially increasing supply of labor in some sectors drives down wages for everyone in those sectors and in the labor markets that compete with those sectors.