The question is very good, and was, famously, discussed by Reb Yisrael Salanter. Coincidentally, yesterday I head what in Telz they call a shiur daas on that topic. Rabbi Levin said in the name of his wife's grandfather, Reb Chatzkel Levenstein of Ponevezh, an answer in the name of Reb Yitzchak Blahzer. He said that only after Klal Yisrael is mekabel ol malchus shamayim is it possible to have a day of kapara. Only after we realize that Hashem is our king, and that by not doing what Hashem wants we are mored bemalchus, then Hashem is willing to be mochel.
I recently saw someone an interesting and relevant mashal that clarifies his idea. A king sends two servants to do some menial job. One is very skilled at this work, and does it perfectly, but feels that is an unfair imposition, and so he does it with resentment and a scowl. The other is not particularly good at this kind of work, but he does it with pride, happy to be able to serve his king. What is the difference between the two? Who is the better eved? The difference is not just who is the better eved. The first one is not an eved at all, he is a moreid bemalchus. The second one is a loyal eved.
So it's not just a matter of doing aveiros. When we do mitzvos, if we do them besimcha, then our actions proclaim that we are avdei Hashem. If we do the mitzvos, but we do them as if they are a burden, then we are moreid bemalchus.
Reb Chatzkel added that there is a difference between a melech and a talmid chacham. A talmid chacham has the right to forgo his kavod; machal ahl kvodo, kvodo machul. A king has no such right; melech shemachal ahl kvodo, ein kvodo machul. But why can't a king be mochel? Because his malchus is not his, it is a stewardship granted by the Ribono shel Olam, and the king can't be mochel on what is not his. The Ribono shel Olam, on the other hand, is a King whose malchus belongs to Him, and so He can be mochel on His kavod. But the Ribono shel Olam is only willing to be mochel on his kavod for his avadim. First, we have to be Avdei Hashem. Only once Klal Yisrael firmly and proudly declares, and demonstrates through their behavior, that Hashem is their king, only then is Hashem willing to be mochel ahl kvodo. After Rosh Hashanna, when Klal Yisrael is mamlich the Ribono shel Olam, only then Hashem chooses to be mochel ahl kvodo. Only then is the mechilla of Yom Kippur possible.
In fact, we can say that asking for kapara on Rosh Hashanna would be contrary to the idea of Malchus. Why should a melech be mochel? A melech is makpid that his servants do his will with pride and alacrity. Bringing up our failings and asking that they be dismissed diminishes the whole concept of Malchus.
And what is the best way to be makabel ol malchus shamayim? Chazal say that it is through Malchiyos, Zichronos, and Shofros. But I like to say over the Tanchuma in Parshas Tzav, which I think is also key to kabalas ol malchus shamayim:
Perhaps we can apply the Tanchuma to the Gemara in Brachos as well:
מה תעשה לשמך הגדול
מי יחוס עלי חוץ ממך
לכן אם תקטלני לך איחל
אברח ממך אליך
ובשולי רחמיך אחזיק
עד אם רחמתני