The majority of Jews live in Israel and at least the majority of Jews are tamei. The reason of being tamei and not being able to bring the karbon only applies to individuals but not when it is the majority of the Jewish people
Torah Mitzion has an article on this by Rabbi Aviya Rozen. Some extracts from this article (with changes to the unusual transliteration and grammar) are shown below:
Is it possible to sacrifice the Korban Pesach on Har-Habayit, today, without the Beit Mikdash?
The answer is very clear -yes. (according the Halacha and ignoring the political situation)
The Mishna (Eduyot 8,6) says:”R Yoshua said: I have heard that sacrifices may be offered even though there are no Beit-Mikdash,and that the most holy sacrifice may be eaten even though there is no Beit-Mikdash, and lesser sacrifices may be eaten even though there are no walls [of Jerusalem]…”. The Rambam (Beit Habechira 6 , 16) says that this is the Halacha and we are obligated to do it, even today. In summary- according to the Rambam it is permissible just on Har-Habayit although there are problems to go up there.
Most of the objectors thought that it is impossible to sacrifice Korban-Pesach today because we haven’t a Kosher Altar, and there no option to build the altar until that Moshiach will come. The question now is if is it possible to build altar?
Rabbi David Fridman and the Rabbi Shelomo Sakal A (Beit Shelomo Yore-Deia part b ,chapter 125) objected to sacrificing a Korban-Pesach today because we haven’t the ability to build an altar. But is just a technical problem.Today with the modern technology, there is no problem to build an altar with perfectly square corners (maybe more so than in the past). In addition -we can use with laser rays instead of metal tools.
The “Tumah” (impurity)
In the Gemara (Yoma 6b) there is a debate about the question if the impurity will be cancelled (הותרה) or just pushed aside (דחויה) in a case when all Israel are tomei meis, but everybody agree that we can sacrifice a Korban-Pesach in that situation.
The Rambam (HilchotBeit Hamikdash 4, 15-16) says that the Halacha is:"the impurity is pushed aside if all Israel are tomei meis.” In addition -the Rambam wrote that if there are no Kohanim or there are no Jewish person who is not tomei meis, then the Korban can be sacrificed by any Jew even if he is impure. (Rambam Beit Habechira 7,23)
So the issues seem to be:
- The political problem of going onto the Har Habayis.
Whether "all Israel is tomei meis" is satisfied by that proportion of the Jewish people living in Israel.
What proportion of the people are affected by tumas zivoh (for which there is no permission to ignore) - see this question and its answer.