> You might talk about the Persian Empire when referring to the Achaeminids or the Sassanids, but that comes with the understanding that while the ruling class is Persian

not exactly. Persians are also majority of people in what's now called Iran (just like Russians are majority in Russia, this is the same naming pattern as for many countries) and renaming is a result of an invasion. Talking about "ancient Iran" before it Muslims arrived is talking about Persia