Give or take, that works for cis people. Trans people... my understanding is that their mind's gender does not match their body. Now, I technically count as trans (I sure as hell wasn't assigned "nonbinary" at birth), but let's restrict ourselves to binary trans people. So, to use your analogy, instead of being happy being a Hyundai, they want to become a Toyota, but they're a Hyundai physically. Yes, that is probably the stupidest analogy I've ever come up with.
There's a conflict, and the only way to resolve it is to become the opposite gender (again, ignoring nonbinary people; lies to children and all that). This involves a lot of things, from changing clothing and mannerisms to injecting hormones to surgery. It depends on the ability of that person which of those things they do. Some people everything, but some only do a few things. It doesn't make them any less trans because they don't wish to do this or that; there's probably something underlying preventing them from doing so, and some people just don't feel comfortable taking drastic measures.
Of course, it's entirely possible to say that gender is cultural and it shouldn't matter, but it's sorta ingrained, isn't it? From a young age, you're taught: "This is a boy, and this is a girl." So it would make sense, then, when someone doesn't feel like they fit in the box that they've been given, that they'd try to move away from it. Sometimes it's just a bit of a phase (see: r/notlikeothergirls), but often it represents a difference in identity from the "norm" (cis, heterosexual). Whether it's just being gay/bisexual/something else, or if it involves a change of gender (or both! They're not mutually-exclusive.), it depends on the person.
Me, I became agender because... the idea of being a man just didn't sit right with me. It would take 10 years in brain-time to fetch my gender from the giant archive that is my long-term memory. In hindsight, I should've noticed; slow information retrieval is a very strong indicator of reluctance about the topic. But now I've changed, it makes a whole lot more sense to me. Being addressed as nonbinary makes me legitimately happy; I believe the term is "gender euphoria", the opposite of "gender dysphoria".
And that's sort of the thing about trans people. It makes them happy to be the gender they want to be, and how they go about it is dependent on them. Me, I couldn't care less. I'll continue to look the same as I did before. Some, they go through some fairly major changes to be that gender. And in the end, is that not what you and everyone else wants in life? To be happy with who you are? To be comfortable in your own skin? Why ruin someone else's happiness, as long as it's not hurting others?
I'd like to end with this very simple phrase: "Don't be an asshole."