Equality is a broad term. Consider two forms, equality of opportunity, and equality of outcomes.
In equality of opportunity, we remove barriers to merit based competition. Everyone competes on the basis of merit. An example of this would be symphony try-outs behind a curtain, where those reviewing the musician only hear the sounds of music, and can not factor in appearance, gender, preference, etc. Another example is a teacher who has an intermediate randomly assign numbers to students in their class and the tests are then graded by that teacher with only the number as identifier. They are graded purely upon academic content without regard to personal preference.
Most every person will agree that equality of opportunity is a good thing, as it maximizes the pool for merit based competition and gives us the best, and gives us a bias free evaluation against objective metrics.
Then we have equality of outcomes. Here you want the same performance across all races and genders and ethnicities without regard to merit. You give bonus points to those who as a group fall below the average, and you take away points from those as a group perform better than average. In this case you no longer have merit based competition, and you are curving performance without regard for quality. A good example on this are the adjustments universities make to SAT scores for admissions, typically penalizing Asians , and boosting Hispanic and Black scores. Those who are penalized purely because they fall within a group that historically performs well feels discriminated against. Those boosted feel as if their accomplishment is diminished.
The argument for doing so resorts to this notion that environment trumps all, and there are no genetic factors in performance. Pinkerton in The Blank Slate refutes this claim. There clearly are both genetic and environmental factors in play. As a group it is well known in psychological research that men tend to orient more towards things, and women tend to orient more to people, and these sexual preferences in career pursuits shows up even more in highly egalitarian societies like those in Scandinavia where the representation differences between the genders is wider than in less egalitarian societies. This was very surprising, indicating that given full and free choice the sexual differences in career choice maximize.
Thus in an egalitatian society in which we maximize equality of opportunity, we have necessarily a widening of the sexual differences in equality of outcomes. At the core of this is liberty, that people when given free choice may pick career paths that do not result in equal outcomes.
And so we come to the crux of it, to then achieve equality of outcomes, one must restrict liberty of free choice which is viewed as onerous by most of society. To get more women into say physics, you would have to compel them to choose that and deny them say sociology as a major. Nobody likes this. We want to be free to pursue our own careers without such restrictions. One can not blame it on sexism when we see in societies in which barriers to equal competition are removed, people voluntarily choose careers that widen the differences between the sexes.
And so I think there is a fundamental incompatibility between equality of opportunity and equality of outcomes, and if I must pick one over the other, I prefer opportunity because it maximizes liberty and results in color blind merit based competition.
Now when there is evidence of sexism or racism and real throttling of opportunity, one must pursue that. You bring it to light with specific cases of where individuals were denied. One can not conclude from outcome statistics sexism or racism exists, as we see from study the voluntary choices result in real gender differences in representation. You pursue hiring practices that blind the evaluator to things like race, ethnicity, gender etc. And you accept the results of those blind and fair evaluations.