What might be some of the ways GP poster's family managed it?

Pretty much nobody does that in the USA (maybe by getting married? Prob not even that in Trump II), where I am. Come in an a tourist visa, stay over, manage to legalize your stay in a few years and then become a citizen. Nope.

https://en.wikipedia.org/wiki/Mario_Guevara_(journalist) this guy spent about 20 years attempting to do exactly that, and was only actually caught and deported last month. In this case, the way he was attempting to get legal citizenship was by virtue of his now-adult children who he and his wife had on US soil, which makes the children legally natural-born US citizens.