"Lies, damn lies and statistics" - Mark Twain (and others)
The devil is in the details, as often the case. This isn't the 60's or 70's. The article references costs in 2015. There are far more two income families now, which necessitates day-care and summer camps for the kids. The article also says it's cheaper in rural areas, to the tune of 27% cheaper, dropping the annual cost to under $9,500. This is an average, and there are some big dollar spenders, especially in the northeast and California, plus other urban areas.
The biggest portion is housing at 29%. Yes, you may have been crammed into a tiny bedroom with one or more other siblings, but that's less likely today. Still happens, just less likely. Heck the first floor of my house is probably bigger than that home in the 60's and there are no bedrooms on the first floor of my house. My kitchen isn't much smaller than that house (and bigger than many apartments in NYC) and my kids like to tell me about how much bigger some of their friends' homes are.
You didn't have cellphones and laptops for everyone in the house back then either. I don't recall being in activities every day of the week either; we cut back on that quickly as my wife and I were getting worn out taking them to soccer/gymnastics/taekwondo/etc.
The article also references costs getting cheaper with more kids, "For families with three or more children, per child expenses averaged 24% less on each child than on a child in a two-child family."
I just wish the lower income people would get this message and reduce the number of kids they have that the rest of us have to pay for.