A lot might depend on whether we declared a space race on, as OTL, or not. JFK, to manage the Vostok-Gagarin crisis, decided to promise we'd win a Moon race he just made up, and also set a definite time limit for us to accomplish it first by. If we had blown that one, especially if we failed both hurdles--missed the 1969 deadline (or if our politicians were really persuasive, 1970) and then the Russians beat us before we could manage the landing ourselves, we'd be in pretty bad shape I think. Perhaps we'd be in the throes of a cultural revolution of some kind and either bow out of the race on the grounds that our new Enlightenment says that was stupid, or tender the excuse we were distracted and busy.
But if we had said nothing at all, I suppose it depends on how we handle things after Sputnik and Vostok. Would we gradually back away from the arena, telegraphing that we should be loved for aspects other than our technical prowess? Demonstrate the latter in ways obliquely or not at all related to moon landings, such as economical SSTs, functional nuke plants delivering power, if not too cheap to meter, than anyway cheaper than the chemical fuel competition? Should we laugh at space as an expensive rathole to pour money into? If we can sell ourselves on those kinds of attitudes by 1969, then perhaps we can watch a Soviet moon landing with equinanimaty.
And maybe we can convince ourselves we have a different sort of space program that is better because it is more efficient and systematic or something.