This wasn't what I was talking about, but for what it's worth I don't think welfare and migration are exclusive. If your concern is free-riding, then you can require a certain number of working years before a migrant is eligible for certain social services; or have a scheme in which free migration is required to be reciprocal and bundled with a minimum standard for welfare programs. I believe the EU does both.

The phrase "left-winger" is also kind of vague; lefties in America are very specifically left-libertarians and it was specifically that strain of leftism I was referring to. A lot of, say, European lefties are far less internationalist and far more authoritarian for my tastes. There's an argument in here that American lefties haven't ironed out the contradictions in their political ideologies yet. A lot of the support for free migration from American lefties and liberals only started because Trump took the opposite position and did so extremely. But I'd rather that than, say, mainstream American politics' obsession with endianness[0].

Additionally, the welfare state is currently being crushed under the weight of a bunch of retiring Baby Boomers. Which isn't a uniquely American problem. Every rich country with a welfare state has too many old people in the system. That's due to demographic collapse, which is itself a consequence of children being too expensive to raise in a world where effective birth control is easy to access. Immigration can actually make this less of a problem, by adding to the tax base. They don't even have to be highly-skilled immigrants, they just have to be generating taxable income that goes into the welfare system.

The Globalist Nationalist argument is that this is bad because Those People™ are criminal or uncivilized in some way. But that begs the question: where do we find more taxpayers to keep our welfare state solvent? The answer is to take away birth control and hope enough horny teenagers get pregnant to fix the shortfall eventually. Except this doesn't solve the affordability problem[1]. So all those kids are going to be in broken homes or no home at all. This creates a crime wave. In fact, this is exactly why Those People™ from Those Countries™ are so damned scary!

tl;dr The integrity of the welfare state is only tangentially related to the openness of its migration programs, because migrants can be both beneficiaries of and contributors to the system.

[0] In the original sense of endianness being a culture war over what side to break eggs on.

[1] The affordability problem is itself a cluster of related cost diseases endemic to urban centers. More people want to live in the city than there is housing available for them, so prices have to rise until somebody loses. And increasing the housing supply creates further demand for land, construction labor, and materials. That raises cost, meaning that housing has to get more expensive to make new units economic. And as housing gets more expensive, there's less children being born, which means less people working to build that housing, which makes the housing more expensive, so there's less children being born...