Once we landed on the Moon, the Soviets also used radio triangulation to confirm that our transmissions (”That’s one small step for Man”) originated from the Lunar surface, as well.
The Apollo missions also left behind laser reflectors that we still use today to precisely measure the distance of the Earth to the Moon (varies slightly).
Some universities ping those laser reflectors today, as well.
If we had not sent something to the moon and returned it the Russians would have publicly announced that and humiliated us. Then the only possible fake was to send a rocket to the moon, land something, drop off a laser reflector and return to Earth, but then fake having people there. But a week long ride in space was the easiest of all of the parts of the landing.
The Apollo craft carried transponders that could be pinged. Those were used because radar couldn’t see that little thing so far away. The transponder details were in the public domain. Even ham radio operators with 2.4 GHz equipment could ping it.