There's a couple of different things going on in your question, so I'm going to start with big stuff and narrow down. Which means starting with how the author of the thread (which has a bunch of great points that I'll gladly highlight) uses the phrase "American education system."

Education in America is locally controlled. What this means in a practical sense is that every state has its own history and developed its own approach to determining what gets taught in the classroom. In other words, how the Holocaust is taught in Biloxi is different than how it's taught in New York City. In this older answer about textbooks, I get into some of the factors that shaped the different paths.

Which is to say, a straight-forward answer to your question is, No. American education doesn't universalize Holocaust education. Josh's experiences do not reflect a national approach to a particular topic not because he's wrong, but because there's such a high degree of variability across states, it's impossible to generalize. Some states mandate Holocuast education at one or multiple points in K-12, most do not.

However. There are some things that are common in all 50 states, Department of Defense, and territory schools. From an older response about why Barbary pirates aren't taught in American schools.

Education historians who identify themes across American education history use the phrase "grammar of schooling" to describe the things that are adopted in schools across the country despite the fact there is no national education system. These are things such as referring to teachers by gender identifier and their last name, younger children walking single file down the hallways, older children storing their items in lockers, apple motifs, etc. This "grammar" is what identifies an American school as an American school. It's shaped by cultural norms, mostly those from White Anglo-Saxon Protestant cultures. Which isn't to say the norms of school are only seen in American schools - you can see children around the world raise their hand to get the teacher's attention. The act of hand-raising and waiting for permission to speak, though, is shaped by a particular idea of how children should interact with adults. And to put it plainly, it's very much about a style of communication shaped primarily by the comfort levels of white adults, mostly women in the classroom and men in positions of leadership.

So, while Josh's personal experiences cannot be generalized, the idea of teaching the Holocaust HAS become part of the grammar of American schools. In other words, it's misleading to say teachers in all 50 states, etc. teach the Holocaust the same way. It's accurate to say students in all 50 states will experience some form of Holocaust education in their K-12 experience.

Part of what will shape the specifics of that education is a concept known as Americana. From the same older response:

Americana can best be thought of as the packaging of American history and touchstones for the next generation. It's a framework that led to the "Washington and the cherry tree" genre of stories, generations of school children memorizing the preamble to the Constitution, learning Christopher Columbus "discovered" American and mass dislocation and genocide of Indigenous people was simply "manifest destiny", and other broad strokes about what happened on this soil. This simplistic approach to American history was embedded in the texts children read and the way teachers talked about history. ... This meant that the 400th anniversary [of his landing] was everything. Schools across the country were planning celebrations, not because they coordinated, but because celebrations of events related to Americana was something you did in American schools.

The nature of Americana shapes how schools approach Holocaust education. That is, the pedagogical approach has been actively pro-white, which positions American soldiers as the heroes of World War II and while some teachers may do what Josh describes and tell students, "this could happen to you", others focus on the heroes of The Holocaust, those who hid Jewish families or participated in the resistance. Later in his thread he offers, "I think it might be prudent to teach them similarly— encouraging students to be pro-Black American rather than merely anti-Jim Crow, to be pro-Native American rather than merely anti-Trail of Tears, to be pro-Asian American rather than merely anti-Japanese internment, etc. (15/x)"

The approach he proposes would likely make a world of difference stands in contrast to the approach embodied by Americana as it, in effect, puts white Americans at the center of everything. Understanding this can help us better understand why Holocaust education looks different in different parts of the country.

Educators' understanding of who is centered in Americana, that is, who counts as white, has shifted and evolved over time. One of the reasons Christopher Columbus was elevated was related to a shift in Italians moving towards whiteness (more on that here as it relates to schools and immigrants) In some communities, Jewish immigrants were "othered" or seen as outside the protection of whiteness available to other immigrants, such as white German immigrants, etc. However, in other communities, they became - through a combination of factors that I'll defer to others on - part of the Americana narrative, meaning their stories were centered.

In his 2008 book, The Emergence of Holocaust Education in American Schools Thomas Fallace identifies the work of two teachers in Vineland, NJ in the late 1970s as the origin of a Holocaust education movement. The teachers actively sought to center the Jewish experience in their curriculum and how they presented the history of the Holocaust to their students. One of the ways they accomplished this was by using books like Elie Wiesel's Night which remained centered on the experiences of a Jewish person during the war. However, some communities - especially those where Jewish people were still seen as "an other" or non-white - went with books like The Diary of Anne Frank. While Frank's diary is focused on the experiences of a Jewish girl and her family, the pull of Americana is such the teachers could focus on the brave Germans who protected the family.

A quote from the end of Fallace's book, I think, is a good way to wrap up.

To those who wish to increase instruction on the Holocaust, lobbying for state mandates may achieve a rewarding political victory, but this will not necessarily result in the kind of meaningful experience described here by Mr. Reynold’s former student. In fact, mandating the teaching of the Holocaust undermines the entire manner in which the movement emerged. Initially Holocaust education was not something imposed from above, but rather something that was developed from below. The movement began in the hearts and minds of a few ambitious teachers who wanted to use history to transform the lives of their students. It grew out of an environment that allowed highly qualified teachers the freedom to experiment, innovate, research, reflect, and share. The future of meaningful Holocaust education is dependent upon such an environment.