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I don’t know how to do Reddit quotes but in here one of your chief issues seems to be that “dysphoria [as a] prerequisite for being trans becomes circular.” If I’m understanding, you’re saying that the idea of having dysphoria makes you trans and being trans means you have dysphoria is circular reasoning?
That’s just necessary conditions. If you’re a square you have four equal sides and corners and if you have four equal sides and corners you’re a square. There’s nothing circular about a definition. It’s only circular logic if one of the two prerequisites is not necessarily true, like if I said a square is a shape with four sides, so that shape with four sides is a square. OP is essentially arguing that trans-ness be redefined, so their logic is sound.
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Stripped to the studs you seem to be saying that being trans means running away from your old gender, and that— in terms of identity signifier— running to the new gender can be seen as incidental: a byproduct of running from the birth gender.  
The seems odd to me.  Whether someone's transitioning because they're running from something that feels wrong or towards something that feels more correct seems like it shouldn't much matter.    
It seems to me that you're not taking people seriously.  You dismiss people who say they experience euphoria and not dysphoria, claiming they're just lying to themselves, or repressed.  That might be true, but doesn't it seem more likely to you that people simply have a different internal emotional experience than you?  How often do you see people online dismissing the experience of others because they find it illegible based on their experience: {"Why don't you just choose to stop being depressed?"}.  How often do cis people tell you that your trans identity isn't real because they had a phase as a tomboy.  It's good practice to take people at their word {within reason}, and to understand that people's experiences can vary quite a lot from our own.    
RE: knowing euphoria without dysphoria.  This again, seems odd to me.  Surely, you'd know what loving touch felt like even if you'd never been beat. You can find yourself suddenly happy without having first been miserable.  You can discover a new flavor you like more than anything without suddenly realizing that all your gustatory experiences before that were rank torture.  I have a hard time addressing this point because I find the notion fairly confusing, but it feels important to address it, because this seems to be near the heart of things for you.      

It is also medically proven that those who are transgender and experience dysphoria have brain waves that align with the gender they identify as rather than their gender at birth.

Also, just as a point of interest, this actually isn't true in any meaningful way.  Source: phd in neuroscience and keenly interested in that kind of thing.  The science behind such studies is very shoddy, or have lots of qualifiers that mean it doesn't say what people think it does. >

(To your last point) Really? I never knew that. When I did research on it, it was backed up by various studies and proclaimed to be true by experts in that field.
And yes, I do admit in my post that it could be due to my lack of outside experience. Because I experience dysphoria, I cannot understand someone who doesn't. I could very well be trying to rationalize it in a way that makes sense to me and I recognize that, which is why I am here.
I guess the core point no one has explained to me is this: why would someone begin the transition process, should they not experience dysphoria? Does it just seem appealing? I truly do not understand it, and I am trying my best to. >

Seems like central here is what it means to be trans.  What is the transition process?  When does it begin?  Is it social transition?  HRT?  Exploring gender non-conformity?  What counts and what doesnt?  It's a tough question to answer because I don't think the term is well-defined.  
But someone might, say, engage in some gender non-conformity, realize something about it feels fulfilling or authentic, and gradually feel into that sense of authenticity or beauty only to realize they're some degree or another of trans.  Basically, fuck around and find out.   
I also know some nonbinary people who started down that path only to realize that HRT disagreed with them in some way or another, so they desisted and expressed their queerness in less medical ways. >

Someone can be GNC without being transgender. In the same sense, someone can be transgender without medically transitioning, though I believe that would help alleviate the long-term dysphoria. The transition process typically starts social for experimentation (pronouns, sometimes social expression for personal reasons which is where I will say I definitely agree with the points in your second to last paragraph) and usually becomes medical (HRT and surgery), though, as mentioned, not everyone does it in this order or does it at all given specific sets of circumstances.
Trans-ness is really hard to define, but that's why I define it as someone who experiences a disconnect between physical sex and their mind. Transitioning is a choice that may help that dysphoria go away, but it is not the "forced" choice.
This is how I define it, atleast. Or in my mind, that is how it works and typically how it goes for trans folk. My question centers around medically transitioning more than social to start with — why would someone begin the medical transition process if they did not feel dysphoria in the first place? Or, going deeper, why would they question their gender if they had no reason to in the first place? >

Hi, trans person here. I do experience dysphoria, but sincerely disagree with the concept that it is necessary to experience dysphoria to be trans. I did not start experience dysphoria until several years into identifying as trans. The reason i discovered that i am trans was because of experiencing gender euphoria. Even without feeling wrong in my pre-transition body/identity/self, i had a deep-seated sense of feeling right when experiencing things that align with my current identity. Without having ever, to that point, had an issue with the gender i had identified with and had been assigned at birth, when i began doing things that are affiliated with my true identity, i knew immediately that they were more true to myself than i had experienced previously. This is what led to my identifying as trans. And while it is entirely possible to be gender non-conforming without being trans and to enjoy presentation, gender roles, or actions associated with a gender other than the one you identify with, if someone takes enjoying those things to mean 'i may align more with this gender than the one i previously assumed myself to be,' it comes across as very strange to me to be dismissive of that. >

The issue with dysphoria as an approach is that it tends towards a weird kinda medical/psychological approach. It's often defined in a narrow fashion that exclusively refers to body dysphoria, and the structure as a whole lends itself to gatekeeping. Just think about the way the term is positioned, through the lens of diagnosis and such. Whatever our imagined ideal for the term, this is the way the term often operates. Your approach to the term even includes this, talking about medical professionals. 
You can excise these issues, sure, but the outcome is invariably that your claim, that dysphoria is a prerequisite to being trans, becomes circular. Say we remove the medical aspect. No need for a gatekeeper. We expand past body dysphoria to include any sort of feeling that preferences one set of gendered stuff over another. We include euphoria as well, because the state pre-euphoria can be defined as dysphoria in contrast to that state. Moreover, we recognize that an understanding of "dysphoria" is individual to the person. What are we left with?
Well, at the end of the day, not all that much. Tell me, what does our definition of "dysphoria" mean besides, "Any set of mental states or characteristics that incline one towards being trans,"? At which point, yeah, if you have nothing to incline you towards being trans, then you're probably not trans. But that doesn't actually say much. It's literally just saying, "Behavior has causes." Or, "Identity has causes." Y'know, depending on how you're approaching it. You could equally say you need ice cream dysphoria to want ice cream. Or that you need religion dysphoria to be Christian. Or, hell, that you need gender dysphoria to be cisgender. After all, they need some cause to not be trans.
You can take it all back the other direction, try to add caveats to what counts or does not count as dysphoria, but at that point I become skeptical of your core claim, that gender dysphoria is a requirement to be trans. Broadly, either the term is meaningless, and the claim about it is true, or it has some real meaning, and I do not think the claim about it is true.