What details suggest that Carrel’s claims about the immortal cell line were not scientifically sound?

Carrel's claims became unrealistic and "fantastical".

From the text:

At one point he said the cells “would reach a volume greater than that of the solar system.” The Literary Digest reported that the cells could have already “covered the earth,” and a British tabloid said they could “form a rooster … big enough today to cross the Atlantic in a single stride, [a bird] so monstrous that when perched on this mundane sphere, the world, it would look like a weathercock.” A string of best-selling books warned of the dangers of tissue culture: one predicted that 70 percent of babies would soon be grown in culture; another imagined tissue culture producing giant “Negroes” and two-headed toads.