Why do I pray that the Temple be built?

If I may add, this doesn't exactly answer the question, but still: part of the issue is the centrality of the Temple in the Torah and the mitzvot. Nearly a third of the mitzvot in the Torah. Without it, we have a very great lack in the service of God, if only from a narrow technical point of view. It is reasonable to assume that if we were unable to observe many of the mitzvot that are familiar to us today, such as tefillin, tzitzit, taking a lulav, a sukkah, and the like, even then we would pray for their return. One more thing: our service of God today is the private work of each and every one of us. What we lack in the Temple and what is related to it is the work on a national level.

I don't know how national the work in the Temple is. Public sacrifices are of course for the public, but the public is still not really involved in sacrificing them (except for the half-shekel contribution).
Beyond that, it's hard to accept a system of mitzvot that I don't want in itself and don't wish for only as a means to create an atmosphere of public work.
This is also how I feel about praying for the return of mitzvot. As far as I'm concerned, why bother with the sheep of the Drachmana? If it's impossible to keep these mitzvot now, then I have no problem with it. Except for mitzvot that I understand the benefit of.

How long have I waited for such a question,
The Divine Presence brings unity, will there be one law – Who does not agree with me?

We will be more holy and not considered unclean dead (true purity, red cow ashes)
There will be a prophecy.
And in general, the Third Temple, will be in its glory and character and in all its splendor and its gates like gunpowder. https://maagbible.wordpress.com/tag/%d7%99%d7%a9%d7%a2%d7%99%d7%94%d7%95/
There will be peace in the world, there will be no predators, everyone will eat grass. Fruits. There will be blessing. There will be no disputes at all. We will understand the Torah in its entirety. We will live the truth in truth. There will be one kosher.

In my opinion, talking about the Temple today is like explaining to someone who is blind from birth what color is. For someone who can see, there is no problem, and for someone who cannot see, there is no problem.
The parable is of course appropriate if we are all blind from birth, but we have come across tapes from the past that describe the colors and numbers that we once could not see. And there were probably people back then who asked why pray and return to see? What is missing? But for someone who can see, it is clear to him what is missing and poor.

Maybe it's more correct to talk about redemption, about perfection, and there is no person who cannot understand this and see what a world full of evil, misery and pain there is here and when redemption has come all evil will end.
And we will understand where we are running, where this world is running, and why everything that is done here.