I approach this not just from a "class war" point of view, but from a perspective that is against all forms of oppression.
I of course know that it is possible to experience both privilege and oppression because I do. I am oppressed by class structures and capitalism, I am oppressed by Patriarchy, I am oppressed by hetero-patriarchy and cis-supremacy, and I am privileged by settler-colonialism and white-supremacy, and I am privileged by the euro-centrism and the centrality of english, and I am oppressed and privileged by different aspects of ableism.
Many people in the united states experience their lives as intersection of different privileges and oppressions. It is only a very small group of people that are privileged in every way, and a small group of people who are oppressed in every way.
I refuse to sit around and theorize on the drawbacks of being a capitalist. I reject the "mo' money, mo' problems" framework. Capitalist own slaves: that is their freedom. Even if a capitalist thinks they would be "more happy" living in a world based on communes, where property and oppression were abolished, and there were pot lucks and dancing in the streets every night, THAT DOES NOT MEAN THEY ARE OPPRESSED. That means they are privileged now and they would prefer a different way to live. That is a preference, not a struggle for survival.
Anyone can desire the destruction of oppression and capitalism. Anyone can be a meaningful part of this destruction.
I've been trying to wrap my head around your comment: "But it is not only the "oppressed" who are oppressed by capitalism..."
I think what you are saying is that there is a static concept of "the oppressed." (maybe refering to: the working class, or toiling masses or something) And that these "The Oppressed" are not the only oppressed people. If that is what you are saying, then I agree: I recognize many intersecting lines of oppression in our world, however, if what you are saying is that capitalists are oppressed by capitalism too, then you can go fuck yourself right to hell.