Okay, I have raised turkeys on non medicated chick starter, they have turned out fine, I have seen no difference between chick and game bird as far as outcome, this year I went with the chick starter again, they are fine. I would not use layer ever for turkeys.
A simple experiment will very easily disprove your belief that there is no difference between using chick starter and turkey or gamebird starter. Set up 2 separate brooders and put the same age poults in the 2 brooders. Feed the poults in one brooder chick starter and feed the poults in the second brooder turkey or gamebird starter that is at least 28% protein. You will see the difference.
What is your unscientific basis for not wanting to use layer feed for turkeys?
I have been raising turkeys for nearly 30 years. My turkeys use the same run and coop as my chickens. I feed all of the adults a quality 16% layer feed along with free choice oyster shell. My adult turkey hens are doing better since I switched from gamebird maintenance to layer feed about 10 years ago.