I cannot give a definitive halachic answer to your question, but I can bring some sources that will shed light on the matter.First, we mustn't think that the absence of the Temple is a result of the shortcomings of prior generations, and no fault of our own. The Sages taught, "Any generation in which the Temple is not built, it is as if it had been destroyed in their times" (Yerushalmi, Yoma 1a). The same lack of merit resulting in its destruction has resulted in its not being rebuilt. In fact, the Midrash states a frightening outcome of not yearning for the rebuilding of the Temple, "All the communities that fell, it is only because they didn't inquire after and demand the Beit HaMikdash" (Midrash Socher Tov, Shmuel 31). What can we do to demonstrate our interest in, and increase our merit that the Temple be rebuilt? The Rabbis taught, "Those who study about the Temple, it is as if the Temple was built in their days" (Menachot 111a). Accordingly, the main way to enhance our feeling of loss, and also to demonstrate our desire that the Temple be rebuilt, is through study.
Still, the idea of actually going up to the Mount under very specific conditions is not unfounded. Rambam (Beit HaBachira 7:1-7) enumerates several ways in which one is required to show fear and awe for the Temple: One may not enter with his staff, shoes on his feet, his money belt, dust on his feet, nor spit. He may not use the Mount as a short cut and may not go on the Mount except for a mitzvah. One who has finished his service may not leave with his back facing the Temple, rather he must walk backward toe to heel. He concludes by saying, "Even nowadays that the Temple has been destroyed, one is obligated to act as when it was built: He may not enter except in a place where it's permitted to go, he may not sit in the area of the Azara, and may not act frivolously opposite the eastern gate facing the Holy of Holies." It is explicit from Rambam that it is permitted to go on the Mount. The question is, where and under what conditions?
Since there are different levels of holiness in the Temple, and different degrees of impurity affecting a person, where it's permitted or forbidden to go is going to depend on who and where. There are three general areas on the Mount. The innermost, and most holy is the Azara, which includes the Temple building itself and the place of the altar. The next is called the Ezrat Nashim, which is the open courtyard outside the Temple. The least stringent area is called Har HaBayit, which is the open area outside the Temple complex. The consequence of entering any of these areas while in a condition of most types of impurity is extremely serious (1).
However, regarding the impurity resulting from contact or exposure to a dead body (2), even though entering the Azara and Ezrat Nashim is also severe (3), entering the rest of the Mount (the third area) is permitted (Beit HaBachira 7:15, Biat Mikdash 3:4). While this is a surprise to most people, the Mishna (Kelim 1:8) explicitly states that a Tameh Met is allowed on the Mount. In fact, the Bartenura comments that even a dead body itself is allowed to be brought onto the Mount. Tosefot Yom Tov explains that the source of the Bartenura is from the Tosefta on the verse, "and Moshe took the bones of Joseph with him," which explains the verse to mean "with him — into the camp of the Levites", corresponding to the Har HaBayit (see also Tiferet Israel who cites Pesachim 67).
Interestingly, the more stringent forms of impurity pose less of a problem than the less severe Tumat Met. This is because even though the other forms of impurity prevent one from going anywhere on the Mount, one can become purified from them, which would enable a person to go on most parts of the Mount. However, regarding Tumat Met, even though one may enter so
