I think you misunderstand what, exactly, is going on this week.
It’s important to remember that everything we say about Hashem is, at best, a metaphor. It’s axiomatic that we can’t describe Hashem accurately, so we describe Him in ways that make it more tangible to us. The problem with this approach is that it leads to some absurd conclusions if you take the metaphor too literally, this question being a case in point.
As another case wherein this pitfall occurs: take Bamidbar 28:1, which describes the Karbanos Temidim as “My bread.” Can you really think that the Omnipotent Creator of the Universe needs us to feed Him twice a day for Him to survive - and to feed Him a measly lamb for each meal?! The meaning behind this particular metaphor is beyond the scope of this answer; my point is that when describing G-d, you have to use metaphors, and you have to recognize that they’re not accurate.
Back to the Aseres Yimei Teshuvah. We describe Hashem as a Judge, with the team of prosecutors arguing for our death, c”v, and the team of defenders arguing for our life. There’s several books open before Him, some for life, some for death. We daven for ourselves, and we plead that we end up in the Books of Life.
You ask a very strong question: why bother with this whole charade? Hashem knows if we will do Teshuvah, so why hold court to begin with?
Let me try to answer this question by broadening it. Why daven for anything, since Hashem will just give us what’s best for us, anyway? Because taking such an approach completely misses the point of davening.
I’d like to begin with the famous Gemara in Yevamos 64a:
א"ר יצחק מפני מה היו אבותינו עקורים מפני שהקב"ה מתאוה לתפלתן של צדיקים
Says R’ Yitzchak: Why were our forefathers infertile? Because Hashem desires the prayers of the righteous.
At first blush, this is a terrible thought. It almost implies c”v that Hashem is some egotistical maniac who is going to punish people just so he gets what he wants. Since he wants people to cry out to him, he makes them suffer.
Obviously that’s not correct. So what does this Gemara really mean?
Hashem wants to have a relationship with us. That’s the entire purpose of Creation - as the Derech Hashem (ch. 1) puts it: G-d wants (there’s another metaphor) to do good, therefore He created a world and the perfect creation upon which to bestow good. Given that He is the ultimate perfection, what better gift is there than a connection with Him?
This point is borne out by another Gemara about davening, this one in Yoma 76a:
שאלו תלמידיו את רבי שמעון בן יוחי מפני מה לא ירד להם לישראל מן פעם אחת בשנה אמר להם אמשול לכם משל למה הדבר דומה למלך בשר ודם שיש לו בן אחד פסק לו מזונותיו פעם אחת בשנה ולא היה מקביל פני אביו אלא פעם אחת בשנה עמד ופסק מזונותיו בכל יום והיה מקביל פני אביו כל יום אף ישראל מי שיש לו ארבעה וחמשה בנים היה דואג ואומר שמא לא ירד מן למחר ונמצאו כולן מתים ברעב נמצאו כולן מכוונים את לבם לאביהן שבשמים
To summarize this last Gemara: Why did the man fall every day in the Midbar, instead of once a month, or once a year? The Gemara poses a mashal: A king gives his son a stipend once a year. But doing so means he only gets to see his son once a year. By giving him his stipend once a day, the son must come every day. So, too, the Jews: by constantly being concerned for the next day’s food, they will constantly be davening to Hashem.
Hashem doesn’t just punish us randomly, because He’s an egotistical maniac. He punishes us because He loves us. He wants to have a connection with us. We don’t (or at least shouldn’t) daven because we need things, but rather we should daven because we want to connect with Hashem, and the Berachos will follow.
It’s the same deal on the Yamim Noraim, just scaled up a bit. Why bother davening for our life, when He already is going to give us what we have earned? Because He wants us to daven, to connect with Him. And what better time to connect than the time described as “Seek out Hashem when He can be found, call out to Him when He is close” (RH 18a, from Yeshaya 55:6)? The picture of judgement is only a metaphor - after all, is Hashem a human judge who can be swayed?
This is based on the Torah of my Rosh Yeshivah, Rabbi Yochanan Zweig, Shlit”a.