> should prove that the current systems are already being held up by people who do not participate in the welfare system.
Well, yes. If there is a pool of workers who aren't covered by the welfare system then it would work out fine to just let them migrate. Big wins for everyone. Probably works great every time it is tried. And if you're arguing that in practice there is an underclass in the US that isn't getting welfare and that works then sure, easy to see.
But, and I'm just going by vague rumours from reading US political news, there seems to be a significant number of people who would want US citizens covered by a welfare system. Phrases like "Universal" and "Basic Human Right" turn up from time to time. The people arguing against offering everyone in a country general support have lost a lot of arguments in parliaments around the world since ... around the late 1800s with Bismark as I vaguely recall. It comes off as unfair and unreasonable.
Frankly I imagine the US political process will start asking why undocumented migrants aren't getting welfare of some sort fairly soon if it isn't already resolved that they get something. That seems like it'd be in line with the general trends. If they are there to stay they're locals.
How does all this square up with easy, formal migration? In a practical sense? Rough numbers?