The original poster is asking totally the wrong question. The problem isn't his friend's level of "emotion"; it is his friend's utter lack of social intelligence at how he will be perceived in writing such a thing. The answer is not for his friend to become a smoother wordsmith; the answer is for his friend to get out into the world more, and actually develop a functional level of social intelligence.
The intended reader of this letter is a HUMAN, probably a group of them. Each of them can be expected to be a very, very smart human, who will analyze the letter on many different levels, but nevertheless, a human, with emotions, from which the writer would like a particular (favorable) response. My immediate emotional response, as a human, after reading this, is disbelief, annoyance, and disgust. Disbelief at how clueless the writer is, annoyance at how little useful information he provides me, disgust at how little he must have thought about me and what I might want to see in such a letter, and then a bit more annoyance at what a waste of time it was to even read it. This letter is so poor on such a fundamental level, it's hard to wrap my head around the idea it was written by someone seeking entry into a PhD program.
The primary emotions being expressed in the letter are anger at the injustice and victimhood of being denied, and bewilderment at not understanding the reasons why, and then more anger at finding out his prior education, is, apparently "worthless" in the eyes of these institutions.
So, the reader of the letter should admit into their PhD program, an angry victim of rejection, a victim who seems to believe the fault was with the evaluators, not himself, yet does not show any understanding of WHY others rejected him? A candidate with this lack of awareness does not inspire confidence, doesn't seem like a good bet.
And instead of telling us even what area the "very particular (mathematics) problem" is in, we are left to guess? Why doesn't he just say it's a "problem in the area of (some mathnobabble area, I'll say "online reputation management") involving (slightly more particular considerations, I'll say "mathematical models for detecting falsified reviews"). What is he hiding from us by being vague? Maybe he's hiding... that he actually knows NOTHING! Yet, perhaps he knows a great deal. The reader cannot tell anything useful about what he does and does not know in the relevant area.
A socially intelligent writer would have gone to great lengths to find out why his qualifications were perceived to be unsuitable by other institutions, and then calmly but briefly explain it. How could he appear to not know WHY those other institutions rejected him? Was he so lacking in motivation, he did not bother to find out the specific, detailed reasons for his rejection? I submit that much academic work is about figuring out what the problem is that needs to be solved, and then applying thinking and action to solve it. How is it possible to have confidence in a candidate who accepts rejection without clearly figuring out the reasons, what the PROBLEM was? How can this man be trusted to take effective action to identify problems, and then solve them, as part of a a PhD program? To address this issue, he must be able to briefly and calmly explain what the problem was, and preferably why it will not be an issue at the desired institution. For analogous examples from Finance and Economics, "European institutions require an EU-accredited course in Financial Derivative Models, which was neither required nor available at my prior institutions." Or, "European institutions prefer graduates of the Austrian school, while my coursework was centered around the Chicago school". Explaining why the problems other institutions cited won't be an issue at his desired institution is an exercise I will leave to him.
And overall, there is little to no mention of anything substantive he has done, nothing that can be verified; the primary mentions are of the injustice he feels has been done to him by having his degrees ignored and disrespected by other schools he has applied to, schools who are obviously arbitrary, mean-spirited, and just out to get poor little him.
I hope I reach my dreams? What ARE his dreams, maybe a little statement of them would help? His dreams are so vague he can't even briefly state them? And he HOPES he reaches, not he "is determined to do the work to reach", his dreams? But don't just try to be more smooth; invest time, attention, and emotion to figure out what his dreams actually are, so that he can clearly and compellingly express them in a sentence or two. "My dream is to..." (I'll say, "help the online world become a more honest place") "...and being admitted into this program would bring me closer by..." ("being around brilliant mentors who can help me devise and improve mathematical models for detecting patterns of deception and fraud").
Such a letter should be written to show an awareness of, and caring for, the point of view of others. As it stands, it loudly conveys a lack of awareness in those areas. The readers would be likely to conclude the writer is unaware, a bad thing in itself, or, even worse, uncaring.
The writer of the letter would be better advised to take up some hobby or activity, take a year off from school if necessary, to work on and improve his social intelligence. Maybe go to a university and work as a mathematics tutor for students in the areas he has learned, while taking classes or doing other work that involves interacting with people a lot. It should be an area that requires a level of social intelligence skills to be successful, provides opportunity after opportunity to develop them, and provides immediate and direct personal feedback on how well he is doing. A quick list of ideas could include doing commission sales work, working as a bartender, and volunteering to canvass and advocate for political candidates and their campaigns.