Basically correct. The Temple is located now in the City of David....lower down and south. The western wall is actually part of the Antonia Fortress.
On my Thread, "THIRD TEMPLE NEWS" (I should have added
From Israel) to the title, I RECENTLY, found SEVERAL picture Inside the Stable, and Several Pictures inside the so called Temple Mount in the Southeast Corner. It reeks from ROMAN OCCUPATION with a FULL LEGION, from DAY ONE. I also found ROMAN X LEGION on Several old relics in Museums in Jerusalem. NO INFO ON WHERE THEY FOUND THEM, but the shear number ROMAN X LEGIONS with 6000 Troops, 4000 Support Personnel (shop keepers), and
300 Cavalry, WHERE THERE IN FORCE.

The relics include
LXF stamped on every tile Square, which stand for
Legio X Fretensis, which is what they the Roman X Legion, called themselves. Those Tile Squares were usen in Swimming Pools, Walls, and Floors. On one Woman's Broach, on Rings that had been lost, etc.
Here is a Quote from the Historian Josephus that says it all.
Quote:
Flavius Josephus
War of the Jews
Book 5
Chapter 5
Section 8
. . .
all
the space of the tower of Antonia itself was built upon, to the height of forty cubits.
The inward parts had the largeness and form of a palace, it being parted into all kinds of rooms and other conveniences, such as courts, and places for bathing, and broad spaces for camps; insomuch that, by having all conveniences that cities wanted, it might seem to be composed of several cities, but by its magnificence it seemed a palace. And as the entire structure resembled that of a tower,
it contained also four other distinct towers at its four corners; whereof the others were but fifty cubits high;
whereas that which lay upon the southeast corner was seventy cubits high, that from thence the whole temple might be viewed; but on the corner where it joined to the two cloisters {Covered Walkways down to the lower Hill, one went to the North Gate, the other to the East Gate.} of the temple, it had passages down to them both, through which the guard
(for there always lay in this tower a Roman legion) went several ways among the cloisters, with their arms, on the Jewish festivals, in order to watch the people, that they might not there attempt to make any innovations;
for the temple was a fortress that guarded the city, as was the tower of Antonia a guard to the temple; and in that tower were the guards of those three (14). There was also a peculiar fortress belonging to the upper city, which was Herod's palace; but for the hill Bezetha, it was divided from the tower Antonia, as we have already told you; and
as that hill on which the tower of Antonia stood was the highest of these three,
https://www.biblestudytools.com/history/flavius-josephus/war-of-the-jews/book-5/chapter-5.html
:END QUOTE.
OBVIOUSLY THE TEMPLE WAS ON A LOWER HILL. Why are all the Shops the Troops and Support Personnel, were in one place, Fortress Antonia? It was ROMAN LAW, that the TROOPS and Support Personnel COULD NOT SPEND A SINGLE COIN IN A JEWISH SHOP. So there were a great number of SHOPS ran by Support Personnel, INSIDE Fortress Antonia.
I stitched this drawing from several sources, but the Cloisters (Cover Walkways) I had not found out about them at the time I stitched these together.
