People are inherently dialectic. We are complex beings that could look at the world different ways. And sometimes those ways give what seem to be conflicting answers, and yet each answer is true. For example, the impersonal philosopher's G-d, the view of G-d we are working from when we discuss Infinity, Perfection, Total Unity, etc... Then there is the very Personal G-d we find in Tanakh, Who describes Himself as angry, feeling regret, etc... And even though we know they are idioms, metaphorically used, this is still a call to relate to Hashem as a Him, not as the unfathomable, all-too-beyond-us, Infinite (Ein Sof).
Rav JB Soloveitchik gives an understanding that dovetails well with R' Hirsch's. (See R Yisroel Meir Vogel's answer.) He explains that the essence of tefillah is to change the self, also by noting the "hitpa'el", reflexive, conjugation for "hispalel".
What was best for the pre-prayer me may not be the same as for the post-prayer me, and therefore prayer could change the outcome.
However, exploring how requests might work despite Hashem always doing what's best for everyone involved either way is different than exploring why we daven and what we're thinking about when we make those requests.
The question of why daven is framed in terms of the Philosophical G-d -- expecting results doesn't fit that way of viewing the Divine.
How to daven, however, is specifically about the Personal G-d. With whom I can have a complex relationship that could be likened to that from (1) lover to Beloved -- Shir haShirim style, (2) parent to Father, or (3) subject to King. As the poet put it, "יְדִיד נֶפֶשׁ אָב הָרַחֲמָן, מְשֹׁךְ עַבְדָּךְ אֶל רְצוֹנָךְ -- Beloved of the soul, (2) empathetic Father, (3) draw Your servant toward Your Will..."
(1) Beloved: I have little to say about praying to G-d as Beloved. Not because there is nothing to be said, but because I can only write about what I know. At my stage of spiritual development, relating to "Avinu Malkeinu -- Our Father, our King" is challenge enough.
(2) Parent: When I am having a hard time, I talk to my parents. Not because I think they can necessarily help, but because turning to one's support system in time of need is a natural part of a relationship.
Turning to Hashem is an end in itself, not a means to get what I want. (This last point is a major theme in R' Soloveitchik's writings on prayer.) It I happen to thereby be changed in a way that does mean that whatever kept me from getting what I wanted is no longer true, great. But that's a side effect, not the point.
(3) King: Well, as a 21st cent American, the notion of King doesn't quite speak to me viscerally either. But I have worked in a large corporation, and dealt with a CEO whose decisions impact 10s of thousands, something that involves a little fear / awe of his power over my fate/ so I modernized the imagery to that.
The CEO's assistant provides a desired itinerary for the meeting, but it's up to me to provide the specific substance. But Hashem and I are the active partners in BMBW LLP (Better Me, Better World), and we both want the same thing -- more good in the world in general, and in my corner of it in particular. Therefore, when I ask for health, it is more "wouldn't it be great if You could", and not "why haven't you yet, please..." Back in the work metaphor, "the employees and customers still need that widget, any chance we can provide it yet"?
And like image of Hashem-as-Father, who knows, He might decide ... "Okay, now that you spent enough time reminding yourself that you need Me to get it, giving it to you won't inflate your ego,won't lead you to take it for granted, it would further motivate you to use it as it needs to be used, or something else." But even if not, my job is the meeting with the CEO and to spend time discussing our joint concern.
