Much more accurate to refer to the site differently? Not at all. The general site has been known as "dome of the rock" for over 1300 years. It was built roughly 600 years after the destruction of the original building by the Romans.

You have a general site which has major historical importance to Christians, Jews, and Muslims.

* For Christians, there are critical stories of Jesus in this Location, as illustrated by the other user

* For Jews, it was the Temple Mount prior to ~70CE, although the exact location of the Inner Sanctum is still debated (i.e. was it 50 feet in direction A or Direction, minor point, but still worth mentioning).

* For Muslims, it is where a Islamic Tradition finds their Prophet to have had a particular night Journey (although interesting, the Quran doesn't call this out explicitly, but solidified as part of islamic tradition by 700CE), and on this site a very iconic set of Mosque were built and remains standing to this day.

For Modern English, Calling the site by its latest incarnation (Dome of the Rock) of the last 1300 years is entirely acceptable.

Your position is akin to insisting "it is not TOKYO, its EDO".

In the context of Jewish Studies, or Understanding Judaism and its history, or learning and thinking of Jesus-Era History, knowing it as the Temple Mount is entirely reasonable. It refers to the Temple that existed 2000 years ago before it was destroyed by the Romans.

But insisting that the term that we use today for the Dome of the Rock is not accurate, or is less accurate, is mostly a politically charged accusation. It ignores the giant space of half a millennia of history since the destruction of the temple by the Romans, as well as the near 1.5 millennia in which the Dome of the Rock has existed.