The Moon is only a few thousand miles away

"The Moon is only about 3,000-4,000 miles (5,000-6,000 km) away - close enough to be a projection or a fake."

The Moon is ~385,000 km away - about 60 Earth diameters. We do not trust any one agency for this: laser pulses are bounced off retroreflectors left on the lunar surface by Apollo 11, 14 and 15, by the Soviet Lunokhod 1 and 2 rovers, and by India's Chandrayaan-3, and the round-trip time (about 2.5 seconds) is measured with millimetre precision by observatories in the US, France, Japan, India and China.

Lunar laser ranging: average distance 385,000.6 km (centre-to-centre), measured to millimetre precision. Apollo returned 382 kg (842 lb) of Moon rock (2,200 samples); the Soviet Luna probes returned 301 g; China's Chang'e 5 and 6 returned a further ~3.7 kg. Independent nations, independent instruments, same number.