But ...
GEMARA: The mishna teaches: If after the unintentional murderer’s verdict was decided and he was sentenced to exile, the High Priest died, the unintentional murderer is not exiled. The Gemara asks:
What is the reason for this?
Abaye says: It is derived through
an a fortiori inference:
If one who was already exiled now emerges with the death of this High Priest, with regard to
one who was not yet
exiled, is it not right that he should not be exiled? The Gemara rejects this reasoning:
And perhaps with regard to
this one,
who was already
exiled, his sin
was atoned for by his exile, and therefore the death of the High Priest facilitates his return, but
that one,
who was not yet
exiled, no, his sin was not atoned for and the death of the High Priest should not prevent his exile. The Gemara rebuts:
Is it his
exile that
atones for his sin?
It is the death of the High Priest that atones for his sin, and the High Priest died.
- Makkot 11b
So, not sacrifice/absolution, but death/atonement.
Not the same thing. The Gemara you posted does not teach what the OP stated that
"sacrifice is needed in order to absolve sins from mankind." Further, death/atonement also not needed for "mankind" according to Jewish tradition. That as a concept is not what is addressed in the Mishnah you posted.
This gemara you posted is about unintentional murderer where the verdict was decided and he was sentenced to exile. The statement, "for his sin
s" does not show up in the text.
דף יא,ב גמרא מ"ט אמר אביי ק"ו ומה מי שגלה כבר יצא עכשיו מי שלא גלה אינו דין שלא יגלה ודלמא האי דגלה איכפר ליה האי דלא גלה לא מידי גלות קא מכפרא מיתת כהן הוא דמכפרא
The type of (כפרא) that it is talking about him not being exiled because due to the death of the Kohen HaGadol and not for the sake of "sins" plural. The exile is in part due to the Goel Hadam (the family member of the accidentally murdered person) and also as a type of punishment. Further, the (כפרא) being discussed concerns the punishment of exile for the particular aveira of the unintentional murder, not other aveiroth and not for the "sins of mankind."
Also, according to the Rambam (Mishnah Torah - HIlchoth Teshuva 1:1-3) actual atonement only comes when the person realizes their mistake, makes corrective action, admits them, and does not return to them - then they bring a Qorban if there is Mikdesh.
The Rambam, in the Moreh HaNevuchim 3:40 explains that death of the Kohen Gadol is simply a part of the process of taking away the the rage of the Goel Hadam due to the loss of the Kohen Gadol being a national tragedy and loss - i.e. Goel Hadam will lose interest due to the greater loss.
Further, Rashi states "As he causes the Divine Presence to reside in Israel, and lengthens their lives, while the murderer causes the Divine Presence to leave Israel, and shortens their lives, he (the murderer) is unworthy of being in the presence of the Kohein Gadol. (Thus, he must remain confined in exile until the passing of the Kohein Gadol. - Sifri, 20.)"