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US Republicans, in particular, love to quote MLK out of context, cherry picking from his words in ways that purport to support their dogma.

"Judged not by the colour of their skin" is a line that gets thrown around a lot by right wingers (and, of course, "enlightened centrists"), primarily in the context of their hatred of affirmative action, diversity, representation, or anything that even sounds vaguely like those things (and yes, those are even their own complex can of worms, but they're not interested in having a nuanced conversation about that). Suppose, say, that a TV show decides to cast a non-white actor for a role that's typically been imagined in the popular consciousness as white. Like, say, a Lord of The Rings adaptation decides that Aragorn should be played by a PoC actor, because they specifically want to make the story feel more inclusive to non-white viewers. There's nothing wrong with this as a casting decision, it doesn't break the story, but of course a lot of racists would get very upset about it. Then, when challenged on their obvious racism, they will declare that actually it's the people casting the show who were being racist by only considering non-white actors for the role, and that MLK would be ashamed of these people, after all, they're supposed to be judging by the content of the character not the colour of the skin.

Their use of this particular line sits firmly in the realms of "I don't see colour." It's a thing that people say either a) as a deflection from examining their own racism, or b) in well meaning naivete. It fails to understand that while race should not be something that divides people, it absolutely is something that divides people. You can't just close your eyes and ignore it, because the moment you do you're ignoring the much larger systemic issues that affect people of colour. A white person choosing to personally inhabit a post-race world does nothing for the non-white people who live in the real world where race, even though it shouldn't, is very much a thing that matters.

Anti-racism is hard work. It involves both constantly examining your attitudes and the ways in which your behaviour is subconsciously influenced by attitudes to race, and working to dismantle massive self-sustaining systems of oppression that have existed longer than any of us have been alive. It's the work of inheriting a broken world and having to accept the responsibility for making it better. But the people who don't want to do that work would much rather have you believe that the solution to every problem of race is contained within a simple mantra of personal responsibility, that basically as long as you're not a card carrying member of the KKK you're all good and your responsibility ends there.

This is a huge and complicated topic (that's kind of the point of the OP), and I'm inevitably oversimplifying in my attempt to explain, but I hope that gives you some idea of whats going on here.

I can't even tell what the misinterpretation being rebutted is. Early on it sounded like the misinterpretation was people in general vs his children, but that doesn't make sense because "umm akshually MLK only wanted equality for his children" is not a good look. Then later on it sounded like it was present vs future, but that's not so much a misinterpretation as a wrong answer to the question of "has MLK's dream been achieved yet?"