Not just inspire, but push the envelope. The fallout of traveling to space not only unified the country but provided new technologies which simply had no need to be created until something as epic as landing on an orbiting satellite was attempted. These actions pay dividends unrelated to the mission at hand.
By 'orbiting satellite' I'm guessing you meant the Moon, but I've personally always been most amazed not by the moon landing per se, but the subsequent return from the surface to dock with the waiting CSM.
The Lunar Module is an amazing piece of engineering! It can land, be the base for excursions for more than three days, becomes and impromptu rocket launch pad and is eminently dockable. All that capability fits in a cube with a five meter edge and weighs less than 15 tons, it was developed and built in well less than ten years and it worked every single time.
Some like the raw power of the Saturn V, I think the multi-talented Lunar Module is much more amazing.
