The article doesn't try to justify the exact number of decimal places - and, by eye, the arguments it uses are likely to work to 14 decimal places as well, since the error would be similarly small.
Instead, it tries to answer the thrust of the prompt question: given the massive numbers used in spaceflight, is pi calculated to the greatest possible practical accuracy? Going over the history of the 15-digit version would divert from the interesting part of the article (the effect of precision on calculation) and dilute a nice teachable moment.
Though that fact about the Apollo Computer would make an interesting part of a follow-up.
The Apollo Guidance Computer wasn't 15 decimal places; it was 15 bits. The point is that they didn't use power-of-two word sizes back then; they used whatever word size fit the problem, even though that now seems bizarre.
But the number in the article is to 15 decimal places. Pointing out that that precision comes from the size of the modern double-precision floating point representation doesn't really answer why that representation is enough.
It's a fair point. They would use more decimal places than that if it were necessary, regardless of whatever a double-precision floating point does. Since it's not necessary, the double is adequate (and already widely available by default across programming languages and system architectures).
Instead, it tries to answer the thrust of the prompt question: given the massive numbers used in spaceflight, is pi calculated to the greatest possible practical accuracy? Going over the history of the 15-digit version would divert from the interesting part of the article (the effect of precision on calculation) and dilute a nice teachable moment.
Though that fact about the Apollo Computer would make an interesting part of a follow-up.