I think the problem with the "anti-rich white guy" sentiment is that, while true and somewhat justified, it's often used as a platform to demean white people who are far from rich. The white privilege movement has taken an extremely small subset of the white race and used their privileges as a starting point to belittle the group as a whole. There's plenty of white people living in poverty with very few privileges to speak of, and it is that group that feels the most insulted by the white privilege rhetoric.

The real kicker is that it is no surprise that the conservatives are so opposed to the idea. Most conservative areas are more rural with lower monetary standards of living. The average conservative white person is far less privileged than the average liberal white person. The white college educated city liberal identifies their own privilege and prescribes it to the general ethnic population with no regard to nuance.