Meh. Governments have techs that they are continuously testing, just coz 60s cars don't seem so advanced doesn't mean governments can't do anything better. What was at stake with the space race was more than bragging rights, of course there gonna throw all their money on it.
Improbable? Of course not! Those random people found out coz they are smart and special.
What makes me believe it the most is the fact that the moon landing was a race between several countries, biggest countries at the time and somehow conspirators think the USA was able to fake it in such a way that they fooled entire nations to believe them? And somehow now years later some youtube idiots were able to find out? Seems quite improbable.
Improbable? Of course not! Those random people found out coz they are smart and special.