The fact is that many ancient writers did mention Atlantis. Radford, I’m sure, knows this, as shown by his next sentence, which is absolutely correct: “There is simply no evidence from any source that the legends about Atlantis existed before Plato wrote about it.”
The most important of the Atlantis texts outside of Plato is that of Marcellus, who wrote in his lost Ethiopian Histories that the residents of islands in the Atlantic, probably the Canaries, “preserved the memory of the prodigious magnitude of the Atlantic island, as related by their ancestors, and of its governing for many periods all the islands in the Atlantic sea” (trans. Cory's Ancient Fragments). This Marcellus is otherwise unknown, but probably lived around the first century BCE, and should not be confused with the many Romans of that name. His work, however, is lost and the Atlantis lines are preserved only in the fourth century CE neo-Platonist Proclus (On Timaeus 1.1), who believed in Atlantis. He also preserved the testimony of Crantor, the first commentator on Plato, that Atlantis was real. The Greek writer Strabo (Geography 2.3) and Pliny the Elder (Natural History 6.36)—who it must be said wrote in Latin, not Greek—also discussed Atlantis in their works, both apparently believing the island to have been real. These two authors, however, are both dependent on Plato, directly in the case of Pliny and indirectly via Posidonius for Strabo. There is not enough left of Marcellus to judge his dependence on Plato, which, while likely, cannot be proved conclusively. The Jewish philosopher Philo (Eternity of the World 26.141) also referenced Atlantis in the first century CE, though again only summarizing Plato.
There is not a single mention of Atlantis, or anything like it, prior to Plato.
It’s probably also worth clarifying for the record Radford’s characterization of the time between Helena Blavatsky and Edgar Cayce’s Atlantis speculations as “a century” after “the 1800s.” Blavatsky’s most famous Atlantis speculations were published in her Secret Doctrine in 1888, and Cayce’s most important psychic readings on Atlantis began in 1932. By my count, this gives us only 44 years, not a century. The specific Cayce prediction Radford references, about the so-called Bimini road, was given in 1938, exactly 50 years after Secret Doctrine. We can expand the timeframe a bit by going back to Blavatsky’s first brief mentions of Atlantis in Isis Unveiled in 1876 (when she still hesitated about its absolute reality), but this gives us 62 years.
More importantly, however, the two cranks were not entirely independent. According to scholars who study Cayce, the alleged psychic derived his Atlantis predictions largely from Helena Blavatsky herself and Frederic Spenser Oliver, another writer who was in turn inspired and influenced by Blavatsky. (The connection is obvious from repeated ideas, themes, motifs, and specific vocabulary.)
But these are minor quibbles with an otherwise excellent piece on Atlantis, notable for concisely presenting the convoluted history of the continent, something many books have strived and failed to do in an understandable way.