Never heard of Mechanical Turk, hahah for a Dutch person it sounds pretty racist. Turk is what we call Turkish people.
The bird was named after the country. And the country after the ethnic groups who spoke the Turkic languages, which derive from the native term Türk, which goes back at least 1400 years. Changing the spelling and pronunciation changes nothing.
Saying that we called the country after the bird is buying into populist rabble-rousing without doing even five minutes of fact checking.