Under the US Constitution, this is not a distinction. What you're looking for is just "the children's citizenship" without this qualifier that signifies nothing under the law.
The better alternative is to aggressively enforce employment laws against employers. Immigrants come here and stay here to work.
If undocumented workers are finding productive work in an economy with low unemployment then the problem is that the government is not facilitating them gaining legal status.
Employers need to stop taking advantage of undocumented workers at artificially suppressed wages. This has acted like a subsidy keeping these poor business models afloat. This has led us to the situation we are in now, where we've become dependent on undocumented migrants (food production etc), who we are being taking advantage of (lower wages, less rights), and also trying to villanize & deport them (the article above). All simultaneously.
It's possible with careful coordination of industry, legislation, and immagration, we wouldn't be here. But now that we are, we need to either find a way to improve the situation or reverse it.
We should stop letting employers do this, and then we all discover that we still really want to employ immigrants, we should enable that, legally.
It would be a forcing function.
Looking at the most recent DHS yearbook (apples and oranges, but the best I can find so far) at https://ohss.dhs.gov/topics/immigration/yearbook and scaling to match the curve at the ICE stats page, it looks like illegal immigration was way down at the end of Biden's term.
So maybe the influx was already slowed dramatically. I don't think it's possible to stop people from wanting to go to the US, except by making it worse that the places people are leaving. I don't think that's a worthy goal.
This is a classic political tactic to be able to say you want to do something, but never actually have to do it. Politicians use this tactic when they don't actually want to do the thing for some reason. In this case, it's because it's really unpopular with business owners, who like to employ illegal immigrants.
It's perfectly reasonable to say "we need to crack down on employers in conjunction with aggressive border and deportation enforcement". But I'm very skeptical of anyone saying "stopping the flow is step 1". I've heard that story before!