Well, there's certainly a terminological debate about what to call this region, its peoples, and the states that rule it, but I would hesitate to say that it's "unfair" or a "falsification" in the sense of a malicious intent to use either Persia or Iran.

Persia, as you note, properly describes the small province of Fars, the people from said province, and the much-wider language of Persian "Farsi" that predominates in the region. As a name it has primarily found favor by virtue of being the most widely used name for the region, its people, and its language among European speakers for many centuries.

Iran, on the other hand, is the preferred name of the people who live in the region, and has, since 1934, been the official name of the country that largely occupies the historical region of Persia.

The first thing to note though is that these kind of discrepancies are hardly unique. Mesopotamia/Iraq and Constantinople/Istanbul are both examples of a European word being supplanted by the local variant despite the former's (often continued) popularity. But using a smaller province in synecdoche for a wider region isn't unique either. The same could be said for Germania/Deutschland. Or indeed for the use in Farsi of "Inglestan" to mean the entirety of the United Kingdom.

Secondly, I think there's pretty good arguments for both sides, which is why you encounter The Cambridge History of Iran but David Morgan's Medieval Persia.

Third, and this is more anecdotal, but there are also political ramifications involved. I have met several people of Iranian descent who have insisted that they are "Persian" because of objections to the current regime in Iran.

So in sum, yes, "Iran" might be the more proper name for all of the reasons that you mentioned, except that being ethnically Persian is not even close to being the primary consideration for the usage of the phrase when it is used by historians.

Personally, I tend to refer to the historical greater region extending roughly from Mesopotamia to Afghanistan as "Persia" and the state that currently rules that territory as "Iran" but admit that that is entirely idiosyncratic. It's just that I have also yet to encounter anyone who doesn't admit that it's ultimately a matter of personal preference, given the above.

Source wise: The first chapter of David Morgan's Medieval Persia which I referred to above gives a good overview of these issues using many of the same arguments that I have.

The Achaemenids were Persian not Median though the had combined with the Medes. Many invaders of Iran also adopted Persian customs and administrative practices, like the Arab empires.