What was the Halachic base for establishing the inheritance of the Great Sanhedrin Presidency?

Question

In the line of Nesi'im from Hillel to Gamliel IV, the position passed from father to son even if the son wasn't the brightest sage of the Sanhedrin (see R' Gamliel's testimony to R' Yehoshua).

Of course, this contradicts Rambam’s ruling (Hil Sanhedrin, 1.3) that:

The greatest sage among them all is appointed their presiding officer, who also serves as head of the academy. He is designated by the sages as Nasi (president), who occupies the position of our teacher Moses.

So what was the Halachic base for establishing the inheritance of the Great Sanhedrin Presidency?

Answer 1

I think this is following the rule for succession of a king:

לְבִלְתִּ֤י רוּם־לְבָבוֹ֙ מֵֽאֶחָ֔יו וּלְבִלְתִּ֛י ס֥וּר מִן־הַמִּצְוָ֖ה יָמִ֣ין וּשְׂמֹ֑אול לְמַעַן֩ יַאֲרִ֨יךְ יָמִ֧ים עַל־מַמְלַכְתּ֛וֹ ה֥וּא וּבָנָ֖יו בְּקֶ֥רֶב יִשְׂרָאֵֽל׃

Thus he will not act haughtily toward his fellows or deviate from the Instruction to the right or to the left, to the end that he and his descendants may reign long in the midst of Israel.

Rashi: הוא ובניו. מַגִּיד שֶׁאִם בְּנוֹ הָגוּן לְמַלְכוּת הוּא קוֹדֵם לְכָל אָדָם (הוריות י"א):

He and his sons: This tells us that if his son is fit for kingship, he is before all others.

Devarim 17:20

The Gemara Horios 11b clearly understands the position of Nassi to be analogous to kingship, just that it is a limited power.

Answer 2

Similar to what @Mordechai wrote, Rabbi Avraham Orenstein wrote in his book "Hanesi'ut Beyisrael" (The Princehood in Israel), pg. 15:

"בתקופה הראשונה, בזמן שסדרי בית הדין הגדול היו קיימים, היתה האפשרות המעשית לחברי הסנהדרין, לבחור בכל פעם נשיא חדש מתוכם. לא כן היה המצב בתקופה השניה, בשעה שסדרי בית הדין הגדול נתבטלו, והיה בלתי אפשרי לטפל בכל פעם בבחירת נשיאים, ובהתחשב כי הלל היה מזרע דוד שעליו נאמר: "לא יסור שבט מיהודה ומחוקק מבין רגליו" (בראשית מ"ט י'), גמרו אמר, כי הנשיאות תתנחל מעתה אצל הלל, לו ולזרעו אחריו לדורותם."

Translation: "In the first era (of there being a nasi), while the Great Beit Din ran in orderly fashion, it was possible for the members of the Sanhedrin to choose a new nasi every time from among them. Not so during the second era, when the system of the Great Beit Din was canceled (by the Romans), and it became impossible to organize every time the choosing of a new nasi¹, and taking into consideration the fact that Hillel was of the seed of David, about whom it was said: "The scepter shall not depart from Judah, Nor the ruler’s staff from between his feet" (Beresheet 49:10), it was decided that the princehood from now on would rest with Hillel and his descendants for generations."

Therefore, the halacha seems to have been שעת הדחק, they were in a legal/religious bind and that was the agreed upon solution.

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¹ I think he means that to assemble all the members of the Sanhedrin for whatever purpose would have arouse Roman suspicions that this was an attempted uprising.