The definition of gender as "the system which maintains female people in a subordinate position" seems to be quoted from someone else's tweets (as the author linked to Threader) but it's a private post that is only available to the person's followers. Anyhow, versions of that statement—the idea that gender is primarily social and structural and that it is inherently oppressive, especially to women—indeed appear in much feminist analysis, as the author suggests. Including in transgender people's analysis, as trans people can also be—and usually are—feminists, and many trans people deeply understand gender oppression.
I have consistently "passed" in my current gender role for over two decades. I am treated socially according to how people perceive me. Cisgender people can make their best-laid plans about how they are going to treat transgender people, but if they don't read me as transgender, I have escaped their radar. Which I do. Every. Single. Day. My daily existence proves that their system is nonsense. When someone says that trans women, trans men, nonbinary and intersex people should be treated as "third" (fourth, fifth, etc.) genders, they are assuming they can tell who those people are. Often, though, they will perceive a trans person as cis—or a cis person as trans. By enforcing categories they make up in their head, all they end up doing is stigmatizing and penalizing people whose gender appears unclear to them, a reality-organizing system that is inevitably carried out racistly and homophobicly and ultimately just upholds gender norms, an outcome that is "surprising" given that these self-declared feminists also claim that gender is inherently oppressive (as mentioned in my previous paragraph). I put "surprising" in scare-quotes because their blatant hypocrisy is not really surprising to me.
If cis people recognize the binary gender system as oppressive, then why do they feel the need to harass trans people until the trans people become invisible ("pass") within that same oppressive binary system? My conclusion (arrived at by others, too) is that these certain cis people claim to believe that gender is oppressive, but they are saying this to distract from the ways they themselves uphold gender, especially (in this case) by imposing gender norms on trans people. When they say they oppose gender norms, they are really saying the rules should not apply to themselves. They want to wear whatever T-shirt they want to wear, openly identify as gay or straight, not be talked down to at work. But they are very happy to uphold gender norms to pressure trans people to consistently pass as women or men and hide their histories in the closet at the risk of being harassed, discriminated against, or worse.
If you are indeed truly fascinated by such discussions, I recommend checking out the reflections of transgender people themselves—rather than exclusionary arguments that anti-transgender people make about them—because transgender people also write articles and books and will absolutely blow your mind. ;) Real nuance on the meaning of what it means to be transgender can come from reading 50 different transgender authors (each of whom will say something different) as well as the work of cisgender feminists whose thinking is trans-inclusive, after which the fallacies of anti-transgender arguments will easily reveal themselves.
Cis people may find the time to consider the possible validity of anti-trans positions, but as a trans person, I can't entertain such arguments even provisionally. I can unpack their harmfulness, but I don't need to begin by saying Hmm, this person has a good point. If someone's premise denies the factual reality of the way I live and if they argue for my profound social exclusion, there is no reason I'd want to take them seriously. They've probably never (knowingly) shaken hands with anyone like me before. They don't know what they're talking about. Their argument may also be logically invalid, but I don't have to bother rejecting it on that basis, since their argument is also logically unsound and aims for a world in which I can't survive.