Incidentally this is also why everyone working in a nation has to receive these benefits (and any others guaranteed to citizens), otherwise you get migrant workers who suck up this, quite literally in this case, but don't receive healthcare.
Only illegal migrants aren't covered, legal ones are. If you make it available to everyone regardless of their legality to be on the soil you open up a whole bunch of other issues
Canada largely avoids this problem by not allowing a subset of illegal people to exist in the country. Canadians are polite, but working in canada without some sort of legal status is far harder than in the US. They have lots of immigrants, but vanishingly few illegal ones in comparison to the US.
You can either provide social benefits to your country's poor or you can have open borders. If you try to do both at once, you're providing social benefits to the world's poor, and the amount of benefits you can provide from a given tax base falls through the floor.
Your ancestors genocided the whole continent, if anything that argument is against what you advocate. We're not in the 1700s, people can travel more easily, information travels instantly, what worked 300 years ago doesn't necessarily works now.
I'm all for helping people but you cannot import the world's misery and expect it to be smooth, first because it doesn't solve anything, second because it just doesn't work from a simple demographic point of view.
