Quite a lot of false points to unpack here...
First, the so-called 'refugee' loophole exists because the backlog exists, which in turn exists because both Republicans and Democrats (including the Trump administration) have refused to reform the legal system around refugees and adopt the necessary resources to the system that it needs. To put this all on the Democrats and then set up the "Democrats want open borders" strawman is ludicrous. It's sort of like blaming the entirely of the war on terror on Bush and setting up the "all Republicans are warmongers" strawman as if Obama, Hillary, Pelosi, et al's foreign policies didn't exist as well. There's also your bald assertion that the statements made by the refugees are "flimsy", which is directly contradicted by the actual journalist reporting of the conditions of which they're fleeing.
At the end of the day, if you ask a reasonable random person on the street what "open borders" means, he or she is going to say something like "anybody can come to the U.S. and become a citizen with no process". And that as a matter of objective fact is not the same thing at all as demanding that the U.S. follow its own laws on refugees. The two are just not the same.
Second, as stated before, more immigration means more taxpayers, more consumers, more entrepreneurs, and more workers all at once. This increases the labor supply, yes. It also increases the demand for labor. Anyone with even a basic understanding of economics should get that if both the supply and demand of X increases at the same time, then it's not clear necessarily if the price of X is going to go up, go down, or stay the same. It's a matter of looking at the data.
What does the data find? More immigration doesn't mean less work to be around. Wages don't go down. Work either increases or stays about the same while wages either increase or stay about the same.
You can argue against this, but you're arguing against a principle established in scientific study after scientific study. You're basically in the same intellectual position as somebody claiming that evolution never happened or that climate change isn't real. The economy just isn't a fixed pie.
Hell, there's even a Wikipedia page about this fallacy: https://en.wikipedia.org/wiki/fixed_pie_fallacy
Third, you're entirely missing the point about the children analogy. The question is about population growth. If, as it's assumed by the anti-immigration crowd, new people is necessarily a bad thing because new people compete with older people... then it also applies that children above the replacement rate are bad because it means an increased amount of workers that take away the fixed pie of jobs out there.
You might object that children also consume goods, start businesses themselves, etc. But that's exactly the Goddamn point. Immigrants do all that as well. That's why increased immigration grows the economy and leads to more people being overall better off.