Performance in standardized tests is not “tightly bound to wealth.” Most of the Asians at the NYC test-based schools, for example, are lower income:
https://www.theatlantic.com/education/archive/2019/03/stuyve.... It’s true most people at TJ are solidly upper middle class, but so are most families in Fairfax County (the median family income of the whole county is $150,000). It’s not like families at TJ are richer than those at McLean, Great Falls, Langley, etc.
Random lotteries don’t level the playing field, they eliminate opportunities for people to distinguish themselves. Societies that eschew objective metrics don’t just have radically flat mobility, they have rigid social frameworks based on non-objective indicators. Imagine the “holistic admissions” factors colleges use to exclude Asians (“leadership personality” or whatever) applied on a wider scale.