The way it’s often taught today is different. It’s teaching about the history as a way to justify or support calls for differential or remedial treatment in the present. And that has the opposite effect—it reinforces that we’re different, rather than being the same.
This is where Americans should wake up and learn some lessons from the rest of the world. Encouraging people to develop ethnocultural identity is something that has never worked anywhere in the history of the world. The idea that we’ll teach kids to see each other as different, but then assume those differences are all “good, actually” is a fantasy. The only way multi-ethnic societies have ever worked is to suppress identity.
For example, “Han Chinese” would probably be several different ethnic groups if people were being honest. Likewise, “white people” are also several different ethnic groups—you can see the difference between French and German people in their DNA. They’re no more the same than are Bangladeshis and Pakistanis. What has suppressed ethnic strife in America between “white people” is the homogenization of the population and subordination of ethnic identities to a constructed, synthetic identity.
Funny anecdote: I live in a blue state, so they’re trying to teach my daughter about “BIPOC.” She’s the only Bangladeshi in the class, so her teacher gave her a book about a Pakistani girl, thinking she’d be able to relate. And I’m like “you’re not Pakistani. Pakistanis tried to genocide your poppy and grandma in 1971.”
One thing that you definitely can't trace in the DNA is "that group of people tried to genocide my grandparents", but that seems like an important "ethnic group" distinction to you.
This is not to dispute your main point which I take to be that you stop fighting over "ethnic" distinctions by giving people a new unifying identity, but I still find myself thinking that something is lost in the process, even if it is a proven approach.