Do you think men will accept that we are equal just by women claiming it? Historically, that's not worked out.
They measure themselves via competition. If women sit on the sidelines their whole lives, there is no reason for men to consider them.
The point of the arrow contest was to prove capability, not to make men look bad. Like it or not, to be accepted as equal we have to prove it to them or they'll never accept us as equal.
Besides, what's the point of having every opportunity if you're not going to use it?
I see where you're going here, and I agree with the sentiment. That said, how men feel about being out competed by a woman is on them. Women are not responsible for men's feelings. Maybe we also shouldn't be hampering our son's by mollycoddling them when they fail. Perhaps a better course of action would be to encourage them to improve rather than expect women to take a dive in order to spare their feelings. Maybe they'd even be better off being directed toward things they can excel at.
Maybe we should stop saying men are good at some things and women are good at some things. Maybe we should just leave it at everybody has their individual talents. I think that's the way to get both real and meaningful opportunities equally given to everyone.