But couldn't one also ask whether we had gone out there without this ideological competition between the two superpowers after WWII?
I think that the actual wars have been the nasty things about the Cold War and that this very competition has been, if anything, a good force. It made the Soviets become great at chess, it created a benchmark for their economy (become at least as good as we), it made them start shoot things out first ... and it made us catch up in the last area and go the the moon.
Very often something sublime arises out of something mundane or even pathological. Columbus wanted to go to India and stumbled over the new world. Cochrane wanted to make enough money to retire and accidentally started off a new age for humanity. (Of course this is not limited to exploration. Suppose you are horny and go out to get laid, meet a great man or woman and fall deeply in love. Once again something as mundane as a biological drive leads to something sublime, love.)
That's why I believe that when we will go in space again for real, i.e. not merely shoot probes and machines out there, it will once again not occur because we really want to explore space. Let's say when mining or recycling scarce resources becomes more expensive than mining them in space we go out there again and exploration will be more of a residual, an accidental side product ... but after some time it might very well become the main thing.