"That is, the Apollo Program brought the country together in the turbulent 1960s and helped make everyone feel good about the country"
This certainly was not 100% true, as some communities protested against Apollo program [1], even though, overall, the progress it has caused helped eventually those communities as well.
Nevertheless there were kind of bipartisan agreement to push this program which, in current situation of the divide in the society, would be hard to achieve, given some program pitfalls, like the tragic catastrophe of Apollo 1, which killed Virgil Grissom, Edward Higgins White and Roger Chaffee (worth to not forget those people, who sacrificed their lives for humanity progress). Kennedy (and his wife) personality helped here a lot too, with some murky person, like Nixon, or someone featureless like Johnson it would've been more difficult.
Also the claim that 1960s were turbulent is not really correct. In fact, 1970s were really turbulent with the rise of domestic terrorism [2][3].
[1] https://www.theguardian.com/science/2019/jul/14/apollo-11-ci...
[2] https://www.dhs.gov/sites/default/files/publications/OPSR_TP...
[3] https://en.wikipedia.org/wiki/Terrorism_in_the_United_States