We spend far far more on healthcare than space or sea exploration already.

Indeed, the moon landing wasn't the universally-lauded come-together moment people like to portray it as. MLK had been assassinated the previous year and the Vietnam War was in full swing; there were a lot of things on people's minds (and greatly varying opinions on what the government ought to be spending money on). The common memory of the event can mostly be chalked up to a combination of propaganda and the people who tend to be able to get their writing about it published (white Baby Boomers).