Elie, it is great to hear from you again after a very long time!

I am not sure that it is true that it says that you save the life of the Talmid Chacham before the life of the Nasi.

1) The Gemara (13a) states that a Chacham comes before a Jewish king. A Chacham who dies is irreplacable, but if a king dies, almost any Jew could do the job.

2) Afterwards, the Mishnah says that you save the life of a Talmid Chacham before the life of a Kohen Gadol who is not a Talmid Chacham.

3) The Rambam, in his commentary to the Mishnah, writes that the Talmid Chacham comes before the king, but this is only in "Emunah." He writes that the usefulness of the Chacham to the people is very great, but that, in practice, one should not put anything before the honor of the king, even if he is not a scholar, as the Torah says, "You shall surely place the king over you" (Devarim 17:15).

4) Rav Chaim Kanievsky shlit"a, in Derech Emunah, Hilchos Matnos Aniyim (chapter 8, Tziyun Halachah #296), writes that there seems to be support from the Mishnah for what the Rambam writes, because the Mishnah says only that the Talmid Chacham comes before the Kohen Gadol, which implies that he does not come before the king when it comes to practical matters.

5) It seems logical that the Nasi comes before the Kohen Gadol. Therefore, we can say that in the same way that the Mishnah implies that the Chacham does not come before the king, so too he does not come before the Nasi.

6) And this is why the Beraisa, on the first wide line of 13b, tells us that one gives greater honor to the Nasi when he enters the room, in the same way that the Rambam writes that even though the Chacham is more valuable to the people than the king, in practical ways one has to honor the king more.

7) After looking more into this question, I think that it is important to note that the Nasi mentioned in the Sugya is also a Talmid Chacham (even though he may possibly be on a lower level of Chochmah than the Chacham). I saw this stated quite clearly by the Keren Orah (Horayos 13b, end of DH Amar RSB'G). He writes that the Nasi comes before the Chacham. The source is what the Gemara reports, that Raban Shimon ben Gamliel said, "Do they not want to make a Hekera (a recognizable difference) between me and the Chacham (Rebbi Meir) and the head of the Beis Din (Rebbi Nasan)? The Keren Orah writes that certainly in private Raban Shimon ben Gamliel behaved as the Nasi is commanded to and he gave honor to Rebbi Meir and Rebbi Nasan and he loved them. However, among the people, the Nasi (Raban Shimon), who was also clearly a very great Talmid Chacham, must be the first, whom everyone honors. Since the Nasi is the public position of the greatest honor, he comes before everyone else.

(It seems to me that according to the Keren Orah, one saves the life of the Nasi before the Chacham.)

Kol Tuv,

Dovid Bloom