That question frames it in reverse. The right question is whether gender should be culturally important. And I can't think of a single reason why it's good for anyone for cultural roles to be predicated on gender. It hurts men just as much to tell them that they're not supposed to be a nurturing full-time parent as it hurts women to tell them they're not supposed to be a breadwinner.

Now, you can certainly find a few places where men and women tend to have different biological capabilities due to how the body develops during puberty. Given that both height and strength have significantly non-overlapping distributions between men and women, it's unlikely you'll see a woman in the NBA. But that doesn't need to result in a cultural expectation regarding the acceptable gender of an NBA player. It's perfectly acceptable to say "most are men, but it's cool if a woman makes it" instead of "you can't do that, you're a woman."

So as to the question asked in the article? Well, you've got to be a little careful. Men, the individual people to whom the label "man" is applied, are absolutely not culturally redundant. They can participate in culture as much or as little as they desire. But it would be nice if the label "man" stopped carrying unique cultural expectations. Although as Betteridge's law would suggest, that's sure not a description of the present state of the world.