We should be very wary of rhetoric that depends on changing definitions of terms without providing precise definitions (see also "racism"). Put differently, everyone's ideas should be criticized on their own terms, but you oughtn't be taken seriously if you don't even define your own terms (and defining them in terms of other poorly defined terms--e.g., "'anti-racism' opposing racism"--doesn't count).
If you can be convinced that the status quo is oppressive to racial minorities, then it serves to perpetuate white supremacy.
I don't follow your first point, since it seems clear to me that people of color, jewish people, and homosexuals are able to hold white supremacist viewpoints.
This is assuming white privilege is the same as white supremacy, when the term white supremacy has been used for KKK and neo-nazis groups, not mainstream white society since after the civil rights era.
It also assumes that most whites and only whites benefit from white privilege, otherwise it's not so white, and may be more a combination of class, culture and/or historical consequences. Also the fact that white people are still a majority in countries like the US, where a majority in any country likely has similar privileges just by being the majority. One last assumption (in America) is that white culture is a certain way, when in reality the US is primarily an English dominated culture historically, whereas Europe has a lot of cultural variation.
A related issue is that white supremacy is sometimes extended to considering an entire economic system as racist, just because history went a certain way. But there's nothing about an economic system that says any one particular group need benefit more than another.
Crony capitalism creates positive feedback loops where the friends of rich people benefit more than strangers to rich people.
I doesn't take a lot of analysis to see how that can re-enforce the dominance of one race in a society if there's any small amount of inequality to start(1) and people of a given race are mostly associating with others of the same race, since the positive feedback loops in capitalism are significant.
(1) And "small amount of inequality" isn't a fair assumption for the US, where one race started out owning people of the other race.
This isn't racial discrimination or racism or white supremacy, and yielding equal outcomes among racial groups isn't innately desirable. If we assume that all races would be equal today were it not for historical discrimination (quite an assumption given that significant disparities predated first contact between different racial groups and thus racism between them), and we want to correct for that historical discrimination then we can talk about it, but that's fundamentally different than "racism is rampant today" or "we've made little progress since abolition" or "we live in a white supremacist ethnostate" or any of the other left-wing claims.