A very big part about why he didn't identify as socialist, let along a communist, was because he associated communism with atheism, which is a dealbreaker for him. If you listen to his criticisms about the Soviet Union, it always revolved around the perceived lack of freedom of religion. He didn't give a shit about gulags or any other dumb anti-communist talking point. And he wasn't a liberation theologian for the simple reason liberation theology came after and were deeply inspired by him. MLK was just never able to square the circle between socialism and Christianity.