Yes and no.
You can keep your child alive without childcare by your staying home, it just costs you money. Note these are two different goals: looking after my child, and making money. Let's start by saying that in the Western world, there are no legal ways to make having children profitable. It'll always cost you money.
In my case, as I'm a trainer, most of the work is outside business hours anyway - people want to train before or after their 9-5 jobs. So I can be with my children all day, wife comes home and we change over, and then I work. Many couples do this sort of thing, the classic one is a day and night shift couple. In this case, no childcare is necessary and both can work, though less time as a couple doesn't help the marriage. But this sort of shift work's not always possible, so let's assume a couple works more or less the same hours.
Now, paying childcare of $X is, at first glance, financially worth it so long as your day's work earns you at least $X+1. And there are other factors too like not stepping out of the workforce and having some continuity to your career, and so on, where it might not seem a good deal financially today, but is good long-term.
But there are other factors, too. When working at a globogym, I filled up my car every 10 days; now I fill it every 60 days. Each fill is about $50, so my petrol bill has gone from $1,800 to $300 a year, that's $1,500 a year saved. And this also means less maintenance on the car and so on. And being home more means someone can cook, which means less going or ordering out for dinner, and so on. There are quite a lot of savings to having someone at home, whether full or part-time.
For a professional couple this is more so. The sort of people who earn $90k tend to have house cleaners, dry cleaners, gardeners and so on. I mean, this is the original 19th century definition of "middle class": wealthy enough to hire servants. Nowadays not many people have Jeeves making them lobster bisque, but they do have part-time contractor servants in the form of the local dry cleaner or whatever. Having someone at home means you can do away with much of that.
Thus, while there is a financial cost to having a stay-at-home parent (again, this can be part-time rather than full), there are many financial benefits which few consider.
And there are other benefits, too. We see that dual-income families always rushing around to and from childcare are constantly stressed and feeling they're not spending enough time with their kids. So not only do they spend on childcare, they end up spending a lot on ski trips and piano lessons and all that to prove, "Yes Junior I really do love you." And I've seen a lot of kids from high-income families have the "poor little rich kid" thing going on, where they're spoiled and lonely. So you worked very hard and spent a lot of money and now they hate you.
So it's not just about money. But even if it is, in the case of person A earning $75k and person B earning $90k, if one of them stays at home with the kids, it should be A. In this article, that means the man should stay at home. This was not an option presented in the article.