The Irish and the Germans came to America when it was unfarmed and undeveloped. In short to build a new country.
People coming from overpopulated third world countries are coming here to grab a piece of the pie that was developed by generations of others. Why don't they stay in their countries and try to make something of what they have there? Because their countries are fucked up beyond all hope. The analogy of rats leaving a sinking ship seems like a more appropriate comparison than sturdy pioneers who came to America to farm and build cities and industry.
I could counter by saying that wealth was concentrated in the agricultural sector in the 1800's and is now concentrated in the commercial sector and a Syrian guy who opens a pizza shop is the equivalent of a guy from the Rhineland who bought a farm in Missouri in 1855.
Fact is that some countries - i.e. Canada, Australia, the US - encourage immigration.
And the European countries have all signed the UNCHR which mandates allowing refugees to seek asylum in those countries. If newcomers are making refugee claims, then "rats leaving a sinking ship" is a decent analogy, but it doesn't quite help you as much as you thought. Those countries can renounce the UNCHR, as can the USA. No one has. Why is that?.....
Because the rightie, anti refugee pols are far happier having the newcomers as a political issue to exploit than they would be if the issue was legislatively solved. It gives guys like you - and Nigel Farage - something to chomp on and it's useful. Without brown refugees, the UK Tories would never have won the Brexit referendum in 2016 and cemented a disastrous 7 more years in power in GB.