I believe that the area of disbelief would be that we just... stopped.
Unmanned space exploration is amazing, and we've done a ton in LEO, but we haven't put a person out past the Hubble telescope since Apollo 17, which was 1972 if I remember correctly.
"Yeah, we went there a few times, there was nothing there besides a bunch of rocks. We brought some back for study, and spent the next few decades on more obviously productive pursuits. Like putting robots on Mars!"
I think it would depend who they were and what/who they knew. Political will wasn’t all that great for the moon, Kennedy even invited Russia to make it a joint venture multiple times more or less to save face while splitting the costs (in the 60s, but still). If Kennedy hadn’t been assassinated there’s a very good chance it wouldn’t have happened when it did and it would be seen as Kennedy’s folly or something.
So someone in the late 50s who was familiar with the actual feelings around budgets and such, might not be so surprised.