There is a lot going on here.

First of all, they are more or less correct saying that "homophobia" and "transphobia" are concepts rooted in the modern world system. While analogous concepts are not unprecedented historically, the idea of being transgender/sexual or homosexual/queer is a modern definition of certain behaviors and identities. As such, homophobia and transphobia can't exist prior to the identities and categories they are oppressive towards also exist.

There is also an important component here mentioned in the response you quote that these concepts are inherently white and Western. There is an ongoing dialogue about the place of non-Western individuals within the larger dialogue around racism, feminism, and queer theory. There is even a question of how nominally "Western" individuals such as an African American woman fit into the discourse on feminism when they also have to grapple with the dimension of race. This all comes out of post-colonial feminist writing which basically pointed out that modern feminism was based on the experiences of white, Western woman and that maybe the model needs to change to incorporate all women. A similar critique can be applied to queer theory, in that the categories the Western LGBTQ+ movement couches itself in, such as "queer" or "homosexual" are specifically Western concepts and/or identities.

Given all that, the answer is that the original post you quote is both correct and incorrect. Correct in the sense that the modern gender binary in a country like the U.S. or in Europe is culturally specific and restricted to a relatively recent period of time. Incorrect in the sense that there are certainly analogous social concepts in many historical, pre-colonial cultures. However, it is important to note that there are in fact a variety of social conceptions of gender in historical and modern cultures that are non-binary, but it is also important to note that binary systems of gender did also exist historically. See this online journal for a good overview of the issues brought up by post-colonial feminism. Also see this previous thread for more discussion about historical views on homosexuality, and gender generally, which gives a lot of great historical examples of how gender and sexuality were viewed, including binaries.

Where I think your original quote gets it most correct is in the last sentence by saying that to end homophobia and transphobia you also need to end racism, as well as in the the second sentence in stating that the gender binary upholds racism. I don't think you can really argue the only purpose of the gender binary is to uphold racism, but it is certainly a component. Forgive me if you are already familiar with this, but this idea is known as intersectionality (see this article for the first coining of the term). Basically, the idea is that different kinds of oppression (such as homophobia, racism, or misogyny) intersect to create an oppressive social structure. Following from this, you can't understand racism without understanding other forms of oppression.

The best example of how this relates to colonialism and imperialism (as you original quote claims) is the often repeated relationship between white, male colonists and indigenous women. There is a specific pattern of rape and other forms of domination towards these women by colonial men that is different than similar kinds of abuses and oppression against European women. While in a European colonial society it would generally be poorly looked on to severely physically abuse or rape a white woman (though certainly both do happen), the same is not always true of indigenous women. The common factor here, being a woman, is not enough to explain the specific structure of oppression. You have to also understand that one persons status as both a non-white "other" and a woman (arguably another kind of "other") results in the specific forms of oppression they have to deal with.

In summary, I would say that yes, the original post you quote is a misrepresentation of history rooted in political ends, but that it is skirting around very real issues based on a sound understanding of the history of colonialism. The fatal flaw is really in not distinguishing between a modern, Western gender binary that has been perpetuated through European colonialism and other forms of historical binaries.

Edit: Other people should definitely chime in. This is a huge and complex topic and I'm sure I have erred somewhere.