Regarding your point that the height of the rock should be proof that it is not the Even Shesiyah, it is not a proof because we know that the Romans plowed over Har ha'Bayis. In the process of leveling everything, more of the rock became exposed. Thus, the Har was higher back then than it is now.

The Radvaz (Teshuvos) writes that there is no question that the rock in the mosque is the Even Shesiyah. He says it is an ancient tradition, and that as far back as the fourth century, the area of the rock was a place of Tefilah for the Jews.

However, others are in doubt about this. Rav Moshe Sternbuch quotes the opinion of some based on the words of the Arizal who says that is a tradition that the site of the Kodesh Kadoshim will never be built upon. Based on that, some say that the place of the mosque is not the place of the Kodesh Kadoshim. Some say that it is much further south, just opposite the area at which we pray today when we stand at the Kosel. (This might also explain why there is an ancient tradition to Daven at the Kosel.)

However, this opinion is quite implausible based on the Mishnah (Midos 2:1) that discusses the area of the Har ha'Bayis. The Mishnah says that the Har ha'Bayis was 500 by 500 Amos, and it discusses the relative distance between the Azarah and the walls of the Har ha'Bayis (the open area on each side of the actual Beis ha'Mikdash, between the Mikdash and the walls of Har ha'Bayis). The Mishnah says that the greatest distance was to the south, after that was to the east, and then the north and then the west. The area of 500 by 500 Amos which the Mishnah mentions refers to the area of the Har ha'Bayis before the additions of Herod in the north and south were built. We know where those additions are today. The southern addition starts from about the point of the Mograbi Gate. If the Azarah was opposite the Kosel, that means that there was very little open space to the south of the Mikdash.

A third opinion is that of Rav Shlomo Goren in his numerous books on the subject. He suggests that the rock is the place where the Mizbe'ach was, and not where the Kodesh Kadoshim was. If the rock was the place of the Kodesh Kadoshim, then it will come out that there will be more space from the edge of the Azarah to the western wall than to the eastern wall, which is contrary to the Mishnah in Midos cited above (the only way that the space to the western wall will be less than the space to the eastern wall is to measure with the smallest possible sized Amah, and to move the western wall inward towards the rock). Measuring from the western wall to eastern wall today gives us a figure of 500 Amos, according to the size of the Amah calculated by the Chazon Ish. If we keep today's measurements (with the Amah of the Chazon Ish), the rock cannot be the place of the Kodesh Kadoshim, because it would push everything too far to the east. It could, however, be the place of the Mizbe'ach.

Furthermore, there is a small cavern underneath the rock, which is consistent with the Gemara that says that the Kohanim would climb down beneath the Mizbe'ach to clean out the Nisuch ha'Yayin every seventy years. It also fits into the tradition (mentioned by the Radvaz) that it is a place of Tefilah. Also, it could be that if the rock is the place of the Mizbe'ach, then there indeed is nothing built over the actual place of the Kodesh Kadoshim, consistent with the tradition of the Arizal. This is also the opinion of Rabbi Reznick ("The Temple Revisited").