There is some research supporting the link, but not nearly enough: https://www.brookings.edu/articles/how-marriage-and-divorce-... (“A child born to a never-married mother in the bottom fifth of family income is 3 times more likely to stay in the bottom fifth than a child born to a continuously married mother with equally low income.”).
It’s possible that it’s not marriage per se. Both Mormons and Asians socialize children into “do what you’re obliged to do, not what you want to do.” Marriage is one aspect of that—you get married, and you stay married, and what you “want” doesn’t really matter. But it might be that mentality that also drives upward mobility rather than marriage per se.