Your question is reasonable, your reaction to the story may not be.
Should children be encouraged/pressured to fit a specific gender role? No.
Should children be supported if they don't fit into a specific gender role? Yes. Always.
Is "nonbinary" a gender? Sure.
Do hobbies and interests indicate gender? Not particularly. Hobbies are gender neutral, and even though society associates some of them with specific gender, they don't mean you HAVE to be that gender to do them.
Do preferences for certain aesthetics indicate gender? Not particularly. Gender roles aren't the same as gender.
Should you gender a child based on their aesthetics and interests? No, you should gender a child based on their gender. Their gender may, rarely, surprise you.
Should I pressure my child to be nonbinary? Not if they aren't nonbinary.
Should I support a child who says they feel like a boy sometimes or a girl sometimes? Sure!
Don't children change their minds about gender a lot? Sure! It can be fun to explore gender and gender roles, and play pretend games about gender - cisgender adults do this too. It's totally natural. If two girls want to play House, they have to either play as a female couple or someone has to be the daddy - it doesn't necessarily mean that they are lesbians or trans boys; they're just playing with gender. It's fun and gives them empathy.
So what, I should support a child in 'exploring their gender' when I suspect they're just a dumb kid who may be making it up? Hey, little girls often claim they want to "marry their daddies," but you don't scold them or put them in therapy for it. You just know that they'll work out the difference between family love and romantic love later. Let your kid explore the difference between pretend gender and their own gender, and trust that they'll work out the difference later. If they think there is no difference, hopefully they'll come to you and talk about it.
Does it matter how old the child is if you want to pressure them into a specific gender role? No, it's always inappropriate.
Even if it's nonbinary? Even if it's nonbinary.
Does it matter how old the child is if they announce they don't fit into a specific gender role? No, they deserve support anyway.
Is it effective to force a child into the "correct" gender role? No, this is not effective.
But isn't it better for them? I mean, if you get it wrong, in 20 years they'll probably cut off contact with you and post in internet forums about how abusive you were. And if you get it right, you made it seem like there was something "wrong" about being the other gender, and you'll end up with sons who won't buy tampons for their sick wives, etc. So no, it's not particularly better for them and could be very harmful.
But my girl child likes cars and trucks and trousers and definitely isn't a boy! Sure. That's because gender (who you are) isn't the same thing as the gender role you perform (feminine, masculine, androgynous.) Some straight cisgender women are very happy wearing masculine clothes and doing masculine things and taking a masculine role. They're still women. Some trans women are seriously into makeup and homemaking. They're still women. Don't worry about it.
Should intersex children or children with genetic conditions receive surgery, treatment or hormonal interference to "make" them into a specific sex? Probably not - you have a chance of not getting the sex correctly aligned with the child's gender, and then you might have the more complicated process of raising a trans child. Unless there is a medical reason why the child has to be treated or assigned ASAP, it's probably best to put off anything permanent until you (and the child) have more information.
Should you put a child on puberty blockers if they are questioning their gender? It'll depend on you and your child. Puberty blockers aren't permanent. However, puberty can cause physical changes that are permanent. If it's still best to put off anything permanent until you (and the child) have more information, then this might be the right decision.
Do families interviewed in the media always come across in a thoughtful, nuanced way? No, not without extensive training. If most families were interviewed in the media about their parenting choices, and the resulting piece edited to sound juicy and entertaining, the parents would sound like raving lunatics and the Internet would advise calling the police and having their children taken away.
How seriously should I take this story, then? With one (1) grain of salt.
How seriously should I take my own children? I don't know, they're not my children.