Confusion over the best way to apply a principle does not negate the principle.
It's not about applying the principle though, it's people disagreeing about what it means in the first place. We can't even agree on how to measure "equal," for example. And if its truth value changes depending on the situation, it's not really a principle anymore.
And that means men and women shouldn't have access to the same rights, resources, or opportunities? That does not follow.
I'm treating belief in gender equality and advocating for gender equality as two different things; I think believing groups are "equal" is probably the most fraught because you have to find a way of measuring/comparing the groups and then somehow it comes out to a tie.
Also men and women probably shouldn't have access to the same resources; women need tampons and support for pregnancy more than men do, because of obvious differences. So where do we draw the line? If there are biological differences in spacial reasoning or strength, that may mean different treatment in some cases. In which case, the distinction isn't necessarily between "gender equality is good" and "gender equality is bad," but "which specific circumstances merit equal treatment?" The devil's in the details. Because "only discriminate when it makes sense" isn't a hot take, even sexists agree with that. They just think it makes sense in more cases.
A woman doesnt need medical care for a prostrate exam, but both men and women need care. This isnt complicated
Which means there's no longer a clear line between gender equality and non gender equality. In the past people would have said "men and women both contribute to society in different ways, therefore women should have to stay at home and cook." I disagree with that reasoning, but I can't just use the blanket logic "it's always unfair to treat men and women differently" when there are cases when it is fair (like with prostate exams). I think the more appropriate response would have more to do with the specific mechanics of the situation and a more general argument for being allowed to choose a job. Ideally one should be able to argue for why women should have the right to work without mentioning equal treatment at all, the issue isn't inequality so much as an arbitrary restriction of liberty.
I dont think MLK would have agreed such comparisons were appropriate. He believed all humans were equal in dignity.
Well I think some people are clearly more dignified than others. Maybe with a narrow meaning of the term "dignity" this is true but there are many other metrics we could use, like "did they commit genocide?"