I do not think you can blame Attaturk, it is my understanding that he promoted the use of these caps. I suspect that it is much like the WWI boots in the US and German trousers, they were worn until there was nothing left of them. There are a large number of originals in New Zealand, as the Kiwis were second only to the Aussies when it came to souvenirs (I think they will have to agree that the Aussies beat them, as they do not have a German tank that was brought home i a ANZAC's haversack.)
Best
Gus