>>8573650NASA did moon landings. You can actually point a big laser to the reflectors they left there and get the light back. Also the flags are already visible from some satellites that go around the moon. In a some years we should have the technology to see all the remains from the Apollo missions from the Earth (Spoiler: the US flag is already white).
The transmission is another thing. They had the landing rehearsed in a studio, not to transmit it in case they failed to reach the moon, but to transmit it in case they had trouble with space interferance. They also practiced with angles and specific reel times.
As for why they did moon landings then and not now, it's because instead of funding big armies for the cold war they funded instead the Space Race.