The modern left tells the working class white person that it is their ( white peoples ) fault that minority members are disadvantaged in society...
No, in general that's certainly not what the modern left (nor centrists and center-right Democrats) say... though it undoubtedly is what far-right media outlets
claim they say. You're new here (welcome to the forum), but there've been innumerable threads on those issues. For anyone actually interested in understanding, it's pretty well understood that the racial disparities and tensions in America - like so many other things - at their root were always
a divide and rule ploy of the rich and powerful, going back to the earliest days of slavery when dissociating and desensitizing poorly-treated working class European Christians to the even greater brutalities inflicted on slave Africans (many also becoming Christians) was an absolute necessity. The greatest societal division had to be black and white, not rich and poor; it's not as if poor Europeans were getting any benefit from slavery, but by instituting 'white' as a privileged category they gained "
a sort of public and psychological wage" as W. E. B. DuBois described the similar phenomenon of the Reconstruction-era South. Similarly in the 1960s, Lyndon Johnson said while campaigning in Tennessee that "
If you can convince the lowest white man he’s better than the best colored man, he won’t notice you’re picking his pocket. Hell, give him somebody to look down on, and he’ll empty his pockets for you," and in the 1980s Reagan advisor Lee Atwater: "
So you say stuff like forced busing, states' rights and all that stuff. You're getting so abstract now you're talking about cutting taxes, and all these things you're talking about are totally economic things and a byproduct of them is blacks get hurt worse than whites. And subconsciously maybe that is part of it." Atwater, like many others, thought that the racial problems were gradually fading away in the decades following the civil rights era, but as the demographic data showed most of the disparities persisted into this century (in some cases even widened IIRC)... and the rise of Trumpism and cult-like devotion garnered by the racist birther-in-chief showed that it was far more than just economic disparities in play.
Psychological wellbeing is important so when a working class person knows that both sides of politics serve the interests of the wealthy/elite then it comes down to which party aligns better with their values. Most working class Americans under Obama felt as though they were constantly being attacked with all of Obamas 'race relations' horse %$%$ and most people dont like being called a racist every day.
As you say, it's a question of where your values align: Do you think that ongoing racial disparities in educational funding, income, killings by police and so on are something which needs to be addressed... or do you choose to believe that they are merely "horse @#%$" and willingly swallow the propaganda lines that you are being personally blamed for all of those problems? Seeing a black man in the White House was obviously a major turning point for America, and the idiot racist birther conspiracy theories of which Donald Trump was the leading proponent were a rather sickening new low in its political scene.
All poor and working class Americans are getting a raw deal from the under-regulated right-wing capitalist system which both Republican and (to a lesser extent) Democrat politicians largely uphold, an anger which far-right media generally and Trump specifically harnessed in terms of white grievance/nativist rhetoric against the black man in the White House, against immigrants, in his support for monuments to Confederate slavers, in calling peaceful black protestors against police brutality like Kaepernick "sons of bitches" and suggesting they shouldn't be in the country, and so on.
It's still the same trick; still exploiting racial divisions and tensions as a distraction from the rich and powerful harvesting vastly disproportionate wealth from the activities of poor and working class folk. As you imply, it doesn't seem like the Democrats did much to try to fix that actual root problem when they held power, nor since they regained it: They (many of them) were merely on the better side of that divide and rule division, siding with efforts to at least reduce those racial disparities and correct some of the injustices. The question then was where the voters' values lay; with the lesser evil, with trying to fix some injustices even if not the really big ones... or with nativism and white grievance rhetoric, pretending that there were no underlying societal problems, or indeed that white people are the real victims in all this?