Not true. It wasn't easy at all. While granted that no one suspected (or at least openly suggested) fraud, the fact is that the "preponderance" of anthropologists concluded that Piltdown was not a single creature, but a chance association of a human skull and ape jaw. The hoaxer had to engineer a second find (Piltdown II) to quell the critics.
The follow-on to Piltdown is Hesperopithecus, in which a dim-witted goofball named Osborne, who ran the presigious Museum of Natural History in New York, lifted a pig's tooth to the light and testified in rapturous tones that it was obviously a hominid fossil.
Actually I don't think either Osborne or any of the Americans ever claimed it was a hominid (in the human family). They claimed it was a hominoid, specifically an anthropoid ape. It was an Englishman, Elliot Grafton Smith, IIRC, who elevated Hesperopithecus to human ancestry, in text accompanying that picture in the Illustrated London News that all the creationists reproduce (and which Osborne immediately dismissed as unscientific fantasy).
The claim that an ape had been found in the Americas was extraordinary enough in itself. Whatever you may choose to believe about the credulity of scientists, the fact is that there was no way in the world such a claim would ever achieved general acceptance on the basis of a single worn and eroded tooth. Further evidence was necessary, and when the researchers at the American Museum sought it out they uncovered their own error.