In a sense we've already achieved all of this, and then some.
During the Apollo missions, we did send our telecommunications via microwaves to the astronauts and back to Earth. But of course, that purpose was not to relay communications to different parts of the Earth, but instead to actually send communications between the Earth and the Moon. The Moon was the final destination rather than the stopover point.
The article mentions using high-flying planes to augment the Moon relay, to go all over the world. This points out a basic flaw in their plan: that the Moon is not always visible in all parts of the world at the same time. So we've already implemented all of the aspects of their plan and improved upon it by incorporating artificial satellites that are always available all across the Earth. These artificial satellites replace both the Moon and the supplemental planes, from their original plan.
There is also a 2-second travel time from Earth to Moon, so there is a 4 second delay on relayed communications. Might have been acceptable back in '47, but we'd be going nuts if we had to deal with such a delay these days. Makes the Moon as a communications satellite even more useless.
Finally, we are actually using the Moon as a relay device right now. The Apollo astronauts left a mirror on the surface of the Moon when they left. A telescope somewhere now sends laser light upto that mirror and waits for it to come back. It's not being used for communications signalling, but rather it's being used to precisely measure the distance between the Earth and the Moon to inches. The electromagnetic radiation being used isn't microwaves either, it's visible light lasers instead. They also find that despite the intensity of the laser light being sent up, they only receive less than a 1% of the signal back on Earth: again showing how useless the Moon is as a communications satellite.
At that time they didn't even envision building artificial satellites. They thought the only satellite we'd ever have was the one natural one, the Moon.