1. Poverty is endogenous to population, not exogenous. Admitting a flood of poor immigrants to your country makes your country poor. All this stuff about being born on the wrong side is misleading, because the difference between the two sides is not natural resources or something* but the people themselves.
Also your stuff about willing employers and landlords is incomplete, because those people do not keep immigrants in cages. The immigrants impose stiff externalities on other people in the destination country, who are not able to adjust their costs and benefits vis-a-vis immigrants by lowering their wages or raising their rents.
2. Restricting immigration is necessary to avert the destruction of the high-capital-to-worker society which is uniquely conducive to technological progress. Even if mass immigration may please some poor immigrants in the short run, it is 'eating the seed corn.' As commenters have pointed out to you before, virtually all the world population growth in the last two centuries has been enabled by the diffusion of technology from advanced countries to poor ones (and one reason most formerly-poor countries are still poor is that they put nearly all their economic growth into population growth rather than capital accumulation, so they stayed near the Malthusian limit!). Transforming all the advanced countries into poor countries by mass immigration will kill the goose that lays the golden eggs. Anyway, hard-core utilitarianism is a suicide pact; clearly non-adaptive. The moral duty to refrain from harming strangers, which is a form of cooperation (offer to participate in 'mutual altruism') does not extend to a duty to relieve all strangers' opportunity costs of not having been born or invited into the community.
Also, there are diminishing marginal returns to immigration. The first few poor immigrants may enjoy big wage gains over their home-country wages (though higher cost of living in rich countries will mitigate those gains) but as more immigrants arrive to compete down wages and fill all the jobs enabled by the available industrial capital, each new immigrant gains less and less over staying home. (We know for sure there isn't much demand for low-wage workers in rich countries-- low wages==low demand!) It is therefore misleading to suggest that open immigration will relieve much poverty around the world, because only a modest amount of migration will force the marginal gains to zero. Sadly, by that point, the quality of life for citizens of the (formerly) rich countries will have dimished toward poor-country levels. So open borders means economically destroying rich-country citizens to benefit a small percentage of world poor people. Temporarily.
3. You tend to destroy your own credibility when you lie, even by omission or by statistical legerdemain. Poor immigrants pay much less in taxes than they (and their offspring) consume in benefits. This is very well documented (in Europe as well as in the US) and conceded even by (intellectually honest) open-borders advocates, and it has been pointed out to you with links to reliable references many times. The closest you can come to justifying your propaganda is to average (as Julian Simon was wont to do) a few hyper-rich Google-founder-type immigrants in with the millions of low-IQ Mexican peasant illegal aliens. That's dishonest because we don't need open borders to admit math geniuses-- we do that already.
4. The "guest worker" approach doesn't work: (1) even immigrants "ineligible for benefits" collect them. They have children and claim welfare payments and schooling for them. They get sick or injured and go to the E.R.. They file for EITC. (2) Poor immigrants excite the sympathy of nice people, the duplicity of leftist politicians, and the cupidity of businessmen. Every grocer in a neighborhood of poor immigrants funds politicians who work to extend benefits to immigrants because the grocer wants his customers to spend more money and he doesn't mind if that money is taxed away from citizens somewhere else. Every employer of poor immigrants is a big advocate for government subsidies to them because those reduce the wages the employer must pay to maintain his workforce-- it's a matter of socializing costs and privatizing benefits. In our society, the only way to avoid subsidizing poor immigrants is to exclude them from the country.
(Nobody is "forcing" anyone to go and live in Haiti. Your prospective immigrants already live in Haiti. They were born in Haiti. Haiti may be a dump but the Haitians made it that way. Americans have no duty to import Haitians to make America a dump like Haiti. Americans who feel sorry for Haitians can send them money. Hope springs eternal, but experience is the best teacher. More than a century of American experience with Haitians in and out of Haiti suggests that no amount of subsidy improves Haiti because the Hatians themselves squander any resources given to them, and bringing Haitians to the US simply adds mouths to the welfare rolls (and inmates to the jails). The only way to "fix" Haiti-- an approach I oppose completely-- would be conquest and imperial administration. I do not advocate doing that.)
Generally speaking. Oil sheikdoms and so-forth are noise.
*Eventually a disproportionate number of the children of low-quality immigrants become criminals and impose additional stiff costs on citizens.