Privilege is not a counter-argument
You cannot make arguments you dislike go away simply by claiming they reflect privilege. Privilege is not a property of statements, it is a property of individuals. It may very well be that the person making the statement is wrong, doesn't know what they're talking about, and speaks from a position of privileged ignorance. If that is true, a counter-argument must explain how and why the person is wrong, not simply paint them with vague, politically expedient Wrongness.
It is perfectly possible to be wrong when claiming a statement arises from privilege. Someone may not have the privilege you think they do; people, when I reject their ‘no platform’ stances, tell me I'm speaking from the privileged position of someone lacking any mental illness or physical disability. Even if they are privileged in the way you expect, you may be wrong in claiming some statement you disagree with arises from said privilege. It is important to interrogate and challenge ones theories of other people's minds. Telling people that they should take the time to understand why others support the policies they do is fine. It's excellent advice that I recommend everyone follow. You cannot hope to deal productively with someone or convince them of anything without some understanding of how they think. At the same time, someone can understand yet sincerely disagree with another person's point of view. They may think their concerns are valid but disagree with the solution they advocate.
The common left-wing assumption that all people in the same identity group must hold the same opinion on some topic or somehow not be ‘authentic’ in their identity is nothing but frank oppression. Pew research, as of 2017, claims that 38% of women think abortion should be illegal in almost all cases versus 42% of men. This is inconvenient for those who would like to frame anti-abortion laws as something done by men to women. Their response is often to deny any personal agency to any woman who doesn't agree with them. This trend of oppression continues in the idiotic post-election desire to excommunicate any LGBTQ or Latino Trump voters as not really LGBTQ or Latino.
This doesn't mean I think it's impossible to act or vote against ones self interest. I'd have to believe it's impossible to be factually wrong to hold that position. Nor do I reject entirely the notion of false consciousness. Poor, white Americans, in surveys, say they think it likely they will be rich in the future. There are other reasons to think white Americans identity with economic classes above their own. Beliefs about class mobility and the sense that there are good times just around the corner make people vote, to some degree, as the rich person they expect to become, and when good times fail to materialize, anger at taxes may well be driven by the sense of having ones wealth-to-be taken away before the fact. However, false consciousness is a thing about which it is possible to be wrong. Social issues drive more voters than we may want to admit to the right, which includes the large swath of anti-abortion single-issue voters. Anti-immigrant sentiment seems to be driven as much by cultural as economic fears, if not more.
One might argue that someone can be mistaken about their self-interest on social issues (people afraid of other cultures may find they actually enjoy having them around), but given the diversity of interests it's much more difficult to quantify. There are also arguments about what causes people to become socially conservative. Here, economic insecurity, failed expectations, and anything framing the world as a ‘scary’ place are known factors. It might be that left-leaning policies that improve the economy by allowing more immigration, make people safer by ending failed ‘tough on crime’ legislation, and create more normalized exposure to others can make the factors leading to social conservatism disappear.