Which, as GP pointed out, is a social dead end.
If your opportunities don't depend on your parents' achievements, then your children's opportunities don't depend on your achievements. Even in a positive sense.
Which means that probably the most important motivation to achieve anything is gone.
Such a society would crumble to dust really quickly.
And, of course, equal "opportunity" just means discrimination based on factors such as innate ability, rather than inherited wealth. Which means that talent takes the place that wealth has in our society, but the basic mechanisms of power (and the resulting inequalities) remain the same.