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.

Foreigners often get tripped up by Leicester - “Lie-Chester” is a common effort.