This is flat wrong. For better or for worse, the modern "White" identity in the U.S. was heavily promoted by late-19th-c. and early 20th-c. Progressives so that light-skinned Europeans would stop trying to oppress and subjugate one another over their national identity. (To be sure, back in the day, these folks did not care all that much about whether other racialized groups got oppressed; they were quite big on "eugenics" for these groups, for example.)
I'm quite ready to admit that this was probably not an altogether foolproof idea, but now we get to live with the results - millions of people in the U.S. treat "white" as a deep part of their identity, no different than being, e.g, "Irish" for others. Many of them would likely take some offense at being told that they should "be less white", or that they need to denounce their white identity in order to "be less arrogant" and the like. Being in denial about how divisive these racially-connoted messages are is just not productive.