The radical notion behind Damien Chazelle’s remarkable “First Man” is this: Tell the story of mankind’s boldest adventure thus far, the Apollo 11 mission that reached the moon nearly a half-century ago, but tell it through a taciturn, emotionally closed-off hero, Neil Armstrong, the first man to walk on the lunar surface. And pay close attention to this man’s state of mind and spirit, as well as to the mission’s spectacular success, so the story is equally about what it means to be flesh-and-blood human while walking on Earth.

The...