I’ve always perceived the New Kingdom as having been “separated” from the cultures of the Middle and Old Kingdoms. Would Coptic speakers have been able to understand Middle Egyptian speakers?
Compared to how modern Greeks can understand the ancient Greek that was spoken and written 3,000 years ago; how would Egyptians compare?
It's pretty unlikely. Old and Middle Egyptian differ markedly from Late Egyptian, Demotic, and Coptic and would have been scarcely more intelligible to Coptic speakers than Old English is to English speakers today. I wrote more about this in How much did hieroglyphics change over time?
Note that even contemporary dialects of Egyptian could be difficult to understand. As the author of the satirical letter Papyrus Anastasi I wrote,
Your discourses are collected on my tongue and remain fixed on my lips, for they are so confused when heard that no interpreter can unravel them. They are like a Delta man's conversation with a man of Elephantine...