Why wasn't Rabbi Akiva made nasi?

Question

Per @user6591's answer on this question:

See the gemara in Yoma 22b. This was orchestrated in order to ensure the longevity of his kingdom. As Rabbi Yoḥanan said in the name of Rabbi Shimon ben Yehotzadak: One appoints a leader over the community only if he has a box full of creeping animals hanging behind him, i.e., he has something inappropriate in his ancestry that preceded him. Why is that? It is so that if he exhibits a haughty attitude toward the community, one can say to him: Turn and look behind you and be reminded of your humble roots. Sefaria's explanation based on Rashi spells out: This is why David’s kingdom lasted while Saul’s did not, as David descended from a family with problematic ancestry, namely Tamar (see Genesis, chapter 38) and Ruth the Moabite (see Ruth 4:18–22).

If this is so, that it's actually important that a leader have some sort of less-prominent background, why wasn't Rabbi Akiva made a nasi instead of Rabban Gamliel?

In Brachot 27b it says:

Shall we establish Rabbi Akiva in his place? The Sages rejected that option because Rabbi Akiva, who descended from a family of converts, would be vulnerable. Perhaps due to Rabban Gamliel’s resentment he would cause him to be divinely punished as he lacks the merit of his ancestors to protect him.

It sounds like in Yoma that having a poor background is a merit because it humbles the person. So wouldn't this have been enough to protect Rabbi Akiva from Rabban Gamliel? Furthermore, surely Rabban Gamliel knew of many previous leaders who didn't have important "Yichus" – In fact, part of Rabban Gamliel's ancestry was the House of David, no less (exactly what was discussed in Yoma 22b) – why then should this disqualify Rabbi Akiva? Why would he attempt to punish him?

Answer 1

See Maharsha to Yoma there (ד"ה תלויה):

נראה דנקט מאחוריו דלא הוה איצטריך ליה אלא לאשמועינן דלא איירי אלא בדופי שאינו נגלה ומפורסם כ״כ דוגמת אחוריו וכמו דופי ממשפחת האם דומיא שפרש״י דופי של מלכות בית דוד שהיה מצד אם אבל דופי שמצד אב שהוא מפורסם במשפחה הוא גנאי למלכות גם לדין של ד״נ דבעי אב מישראל ופסול ברק אמו מישראל ולזה נמי אפשר דנקט נמי קופה של שרצים שהשרצים אינן מגולין כ״כ בה כמו מצד משפחת אם והיינו נמי שא״ל חזור לאחוריך בפסול שאינו נגלה כ״כ ודו״ק:‏

In short, he says that קופה של שרצים represents some kind of questionable ancestry in the female line (as with David, and hence Rabban Gamliel, where the questions raised about David were because of his descent from Ruth). That kind of thing is generally less well known than a defect in ancestry from the father's side (as with R' Akiva, who was descended from Sisera).

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Also, it's not so much that Rabban Gamliel would hold R' Akiva's ancestry against him (and try to punish him for it). It's rather that if Rabban Gamliel would be offended at R' Akiva's taking over the position, then Heavenly retribution would be called down on R' Akiva, to defend Rabban Gamliel's honor,* and R' Akiva would lack זכות אבות to counteract that.

* In the somewhat similar incident in Bava Metzia 59b, R' Eliezer, who had just been excommunicated, just cried about it, and didn't call for any retribution against Rabban Gamliel or the other Chachamim – and yet there were severe effects in the world afterwards, and in fact Rabban Gamliel almost lost his life. Many other such examples can be cited, including from Berachos 38a (today's daf), where two students who offended Bar Kappara passed away shortly thereafter.