You're talking about a 7% improvement in track & field? Hey, apparently it happens. I have no problem there. But we're talking about a 1000-fold (+100,000%) increase in distance traveled in space. That's a 100,000% improvement, not a 7% improvement!
There's not much to do between sub lunar orbit and the moon. It's not like voyages across the atlantic. You have fishing boats go out 100 miles and come back a bunch of times. Then you have some intrepid explorer go out and land in North America. If there's nothing to do in between, there's nothing to do. There were no intermediate flights across the Altantic - just short flights over water and back, and then Lindberg.
Beamon's long jump was really astounding. It wasn't like the high jump which improved with new techniques, or various swimming records that improved with suit technology. The previous record was 27' 4.75". He made a good jump and people thought he might have been close to 28'. He had hit 29' 2", an almost 2 feet increase in the world record of a simple human performance where increases had typically been a couple inches. This was 7% when even drug-based improvements in track records were between 2-3%. And while he did continue to jump 25' and 26', I'm pretty sure he never jumped over 27' again (and
nobody jumped past 28' for about a decade). If it wasn't on video it would be a difficult one to believe.
PLUS nasa says they destroyed the technology to go to the Moon ("how convenient") and today they talk about the Van Allen belts as being this huge obstacle to manned space exploration -- astronauts were on film saying the furthest we can go is Low Earth Orbit (see the end of the country song video in the OP). Double-U Tee Eff?
The loss of technology has been answered many times on the internet, and even on cathinfo. You must be aware of this, no?
The radiation belts were not a significant issue for a brief trip. It might be a problem if people were going to be in a radiation environment for a long time. A Mars mission, for example, would take a few years, and Mars has little in the way of magnetosphere or atmosphere to protect the surface from radiation. (The ISS is in LEO, within the earth's magnetosphere, which provides protection.)