I think part of the problem is that some white people don't feel privileged (i.e. have had their own hardships) and have a hard time understanding that they are privileged. Saying that they are privileged makes them feel disconnected because that's not their outlook on life. It's a hard problem to solve.
How /dare/ they not feel privileged. Each day I thank the lord for the fact that my mother still suffers back pain from her days factory working to buy my shoes. She should be happy to have those aches and pains. Perhaps you could visit her and explain how privileged she is in case it hasn't sunk in.
A lot don't feel that way because it's simply not true. Identity politics involved in assigning people various levels of assumed 'privilege' are almost infantile in their short sighted simplicity. They work on a grid matrix of gender and ethnicity, only two variables. In reality there are many, many variables which combine to form the advantages one individual had over another.
Just a few examples:
Attractive people are more likely to be offered jobs, and more likely to be promoted.
Tall men are more likely to be promoted than short.
Any form of speech impediment will hold you back in life.
Regional accents will affect how people judge you at work.
Where you were born can affect your long term health.
The quality of the schools you attended, and your potential to learn from them can give you a significant advantage. I'm talking state schools here, not private.
Until identity politics grows up and accepts individuals as individuals, it's a waste of time. All the variables above are things that are outside of the individuals control, they're generic or circumstance. And yet, we live in a world where it's seen to be acceptable, even virtuous, to tell someone they're advantaged purely based on the colour of their skin.
Sorry, I'll shut up now and blend back in with the rest of the white male herd, it's easy enough - we all look the same.