First, I am not saying textile armor was never used by anyone anywhere -- of course it was. The Chinese and Japanese integrated textiles into armor for centuries. It was practical.
Regarding the notion of ancient Greeks using laminated linen cuirasses, there is simply no credible evidence for it. The word linothorax itself is a modern invention. It does not exist in any ancient text.
I've seen the oft repeated statement that Plutarch described Alexander the Great's armor being made of folded linen, but Plutarch lived nearly four centuries after Alexander died. Alexander would have worn linen underneath his armor because most clothes of the time were made of linen.
There is the famous mosaic from Pompeii. The mosaic was created more than two centuries after Alexander's death, and more importantly it was a Roman work of art created for a Roman palace. It depicts Alexander with dark hair and dark eyes. Alexander had blond hair and blue eyes. As for the armor, it is rendered with some textile elements; the cuirass has a mix of dark, pale and bright colors. Taken as a whole, the mosaic is splendid art, but not solid enough to stand as evidence.
There is pottery and ceramics from ancient Greece. The base colors of ancient Greek ceramics were mainly combinations of brown, red-brown, orange-brown, black and yellowish white earth tones. Occasionally and rarely there were elements of blue, red or yellow. With that palette as context, the only way to make artistic detail stand out from the background is via contrast. Against a dark background, most details were rendered in paler shades. Against a pale background, details were rendered in darker shades. Thus, cuirasses were most often depicted in paler shades, undifferentiated in color from flesh and clothing. As is the case with the Pompeii mosaic, it is art but not evidence.
For real evidence, there is one fully intact example of an ancient Greek cuirass of the type most likely worn by Alexander. Most importantly, it dates from the right period and has the right provenance. It is the cuirass of Alexander's father, Philip II, who was assassinated in 336 B.C. --

Aside from a few gold flourishes, Philip's cuirass has a basic design and leather construction. Leather, not linen. Many linothorax enthusiasts point to a claim that Alexander once ordered the burning of used armor after a new shipment of armor had arrived. Like linen, leather can burn. May fake history burn as well.