If the Temple had never been destroyed (in the siege of Jerusalem in 70 AD) what might have become of it? Had it stood, in all of its glory, throughout the centuries, worn only from weather and age, what purpose might it have served now? Would it still be a place of worship for modern day Jews or Muslims? Or would it be another site, where tourists could be guided all throughout the different courts and porches? Or possibly neither of these, and kept as a sacred relic?
First, this is a very hypothetical question, as our Lord predicted that the Temple would be destroyed.
Second, the question you did not ask, is whether the Jews might rebuild the Temple? I heard Rabbi David Wolpe (see his website) say that one day the Temple might and should be re-built and animal sacrifices offered there again, in accordance with the scriptures. I don’t know if he was simply speculating or if he was serious.
The lack of a Temple required a re-invention of Judaism by the rabbis in years following the destruction of the second Temple. In this reinvention, prayers became the substitute for animal sacrifice. This was not a leap of insight, because the same problem confronted the Jews in the Babylonian exile, where it is also thought that the ideas of a rabbinic Judaism and of synagogue worship were necessitated.
It is also a problem of who the Temple priests would be, because “Jews” descend from the tribe of Judah, and it was the tribe of Levi who were the temple caretakers (exodus) and priests (Deuteronomy). The lineage of Aaron, and therefore the genuine Aaronic priesthood has been lost, as far as we can tell.
This is strictly my opinion here: Judaism had sort of a built-in self-destructiveness, because it would be impossible for all Jews around the world to obey the commands to go to the Temple three times a year. What I’m saying is that Judaism in terms of Biblical cultic observances is dead. It doesn’t happen and hasn’t happened for centuries. But, it has been fulfilled in Christ.