The modern state of Israel has done a remarkable job reviving a national identity that's hardly existed since the 1st century, bringing back a number of things once relegated to the ancient world. It would seem only natural that a people dedicated to resurrecting an old civilization would have triumphantly constructed a new version of what was once their most unifying structure, destroyed twice by invaders. I'm not Jewish, so I might be fundamentally misunderstanding this topic, but it seems curious that this wouldn't have been among the bold proposals of Zionist leaders. If it was, why didn't it happen, and if it wasn't, what kept it from being so?
Well, I think this is somewhat a political science question in terms of the motivations. Sacred space, however, is an extraordinarily difficult topic to cover. If you'd like a nice discussion of how sacred space is widely regarded as indivisible and easily becomes a flashpoint, you can take a look at Ron Hassner's article, "To Halve and to Hold: Conflicts over Sacred Space and the Problem of Indivisibility", or his book War on Sacred Grounds.
This problem applies amply here. One needs to recall that a Third Temple, being built on top of the Temple Mount/Haram al-Sharif, would also (in most plans) require the destruction of the Dome of the Rock and the Al Aqsa Mosque. When Israel won the 1967 war and took Jerusalem, there was a brief moment where Israelis asserted control over the Temple Mount, by raising an Israeli flag on the Temple Mount/Haram al Sharif.
Within a few hours and after learning about this, Israeli General Moshe Dayan ordered the flag removed, for fear of igniting a religious war. The reason is likely straightforward: the concept of "outrage" can motivate religious war, and this would be no exception. Just as the destruction of religious sites can lead to outrage (see Operation Blue Star in India) and thus to significant violence, Israel was wary of the possibility of motivating Muslim populations now under its rule to fight harder than ever. Turning the conflict into a religious war would have nasty implications for Israel's fighting. Israel thus committed itself to a status quo where Muslim leaders would retain control of the area and its custodianship, with Israeli supervision in times of strife for security changes. Any change to this is seen as cause for an outbreak of violence and leads to calls of religious war. In fact, the mere rumor of a change to the status quo on the Temple Mount/Haram al-Sharif (including before Israel's existence) has inspired conflict. In 1929, rumors that Jews were planning a takeover of the Western Wall and Temple Mount/Haram al Sharif led to protests which devolved into riots. The riots rapidly spread throughout the Mandate (which was then controlled by the British), leading to over 100 Jewish and 100 Arab deaths (the latter mostly by the British during attempts to stifle the riots), including the Hebron Massacre, which prompted the subsequent evacuation of all 600 or so Jews (many of whom had been there for generations) from Hebron after over 60 were murdered. As an aside, the remaining Jews who did survive (over 400 of them) were mostly saved by Arab neighbors who sheltered them during the riots.
As you can see, these conflicts can and would take significantly dangerous overtones in the past due to the indivisible nature of sacred space to the actors involved. Israel, cognizant of that fact, committed from the start to avoiding any change to the status quo. It already had a non-Jewish (predominantly Muslim) population of about 300,000 out of the total 2.3 million in 1965 (13%) or so in the country, and was thus loathe to ignite any uprising or religious revolt, especially after taking control of the West Bank, Gaza, and Golan Heights with their respective populations. Such a move would have doomed any chance for any future peace as well, which Israel took into account after the war ended.