The space race likely necessitated NASA to show some improvement frequently. Otherwise the Soviet Union would have filled the large gaps between infrequent launches with their incremental successes.
The other thing you have to remember is that back in that era, the various military agencies all had a vested interest in rocket technology. Either for suborbital attack profiles or for orbital reasons like recon satellites (which at one point were assumed to be manned, but that didn't prove required).
NASA wound up giving Congress a way to partially unify some of this. Saturn V obviously isn't an ICBM, but if we have the people and technology to make a man-rated rocket to get to the moon it's pretty safe to assume we can build ICBMs to any specification. The military wasn't thrilled with this early on because it meant rockets that were seen as weapons needed to be designed with huge safety margins.
In the end a sort of uneasy truce arose from this and lead to the Space Shuttle. This was intended to create a civilian program with indefinite access to low earth orbit, servicing military and intelligence needs when required. Once it became apparent this was impossible, Congress gave the DoD the go ahead to resume spending on their own ride to space. This in turn lead to the absolute debacle that was the Titan IV. This lead to the EELV program which gave us Atlas V. By this point the US's capabilities had declined so much the best we could do was strap a US made fuel tank to a bunch of Russian made rockets.
