I programmed on the IBM 360 through the 1970s At my first job I programmed on one of the first commercially available minicomputers - SEL32. Minicomputers is not even a distinction any more. My main objection with the movie was the implication the missions would have been impossible without those women mathematicians. They were talented and good at their jobs but not the only people there capable of understanding orbital mechanics and programming it into the main frame or minicomputer. Orbital mechanics basically started with Newton, then a host of European mathematicians improved on it. Every physics and aerospace engineering student studies it, even back then. Might have even programmed the equations up on an analog computer, maybe even a mainframe digital computer if available. It was a team effort something g barely acknowledged by the movie.
The movie was essentially a biopic about those three particular mathematicians, and not focused on the engineering process except to acknowledge that its upper echeolon was a rarified atmosphere to which blacks and women were not typically admitted.
I personally experienced the struggle for equal treatment during my entire corporate career; and I am a white conservative. Not having worked on the engineering side of tech but rather the thought leadership R&D and marketing sides, and also having grown up post-WW2 immersed in the stereotypes about blacks and the South, I came away from the movie just astonished at what these women had been able to do. Today's social justice ingrates would be out on their asses in no time, or else dragging the entire effort down because nowadays they can't be fired.
OK, Langley Virginia is not the Deep South, but even in the DC area, that degree of daily interaction, not even to mention the acceptance of women as team members in any previously male-dominated domains, was quite unusual in the early 60s.
To your point that many others could have done the job: while that is true, the picture was about what actually happened.
It's possible that someone higher-up could have wanted to make a stand for civil rights at the time by seeking to hire black women (there were also white women in those same Langley positions); but it's also possible that these women were only available to Langley because they were shunned from comparable opportunities in most employment arenas. They could also have been seen as a bargain; at that time, not only were women generally paid less for the same work, but also blacks of the era had grown up conditioned to be submissive to the power structures, and therefore hiring black women may have been seen as less scary to male management than hiring white female or male "human computers", as they were called.
I'm not endorsing any of these attitudes; just relating the kinds of things I witnessed.