"The captain wears a coat."
Translation:Ο καπετάνιος φοράει ένα παλτό.
In common speech I never heard anyone say φοραει ενα παλτο.
all I have ever heard is φοραει παλτο.
Although I was born and raised in New York I was raised in a Greek family and we spoke Greek at home and I lived and studied in Athens for several years.
I noticed you are teaching people to use ενα before the noun when the sentence in English asks to translate " a Noun " coat hat dog or whatever. That is not how people speak though.
The sentence is accepted with or without ένα.
It is true that φοράει παλτό is more common when people talk to each other, but φοράει ένα παλτό is used too (at least at Thessaloniki where I live) and it is preferred in essays as it is considered more formal.