Yes on the first point - it's closer to how the Italians pronounce it. But Parmigiano and Parmesan are not the same word.

The English word "Parmesan" comes from the French word "Parmesan", which is pronounced, you guessed it, with a s/z sound.

The soft g/j is not more accurate, because it's a different word. It's an over-correction.

FWIW, the word was in English nearly a whole century before the first British colony in America. It really is a word that existed and folk in America changed the pronunciation of.

Unironically though, I do think that is how i say it. It's parmesan Luc not parmesan Grey.

All languages and country have all sorts of different historical names for foreign places or things. And that is fine. Except when Americans do it. Well and Germans calling Barcelona "Bazelona". That's just not okay.