I think MLK is revered as a symbol more than as a person. The same is true of virtually all of our heroes, who "personify" a value more than reflect the actual person.
MLK rightly challenged this nation on whether core values expressed in the Declaration of Independence was just empty words or the fundamental values on which our society was built.
In essence, as long as any group of citizens did not enjoy all the rights of citizenship, our country was living a lie. And so, MLK stands as "the symbol" for equality of citizenship and calling out inequality wherever it exists. Is that what he believed? Not sure it matters.
Interestingly, MLK did not suggest we not "judge" one another, but rather we do so on the basis of character not skin color. Equal rights, but not equal outcomes.
It should also be noted, that because MLK has been made a hero and symbol does not mean as you suggest that most people agree with everything he stood for. I love what Jefferson wrote in the Declaration of Independence and yet this same person who declared that all men were created equal owned slaves. I clearly don't admire that about him.
I wonder how MLK would have responded to progressive liberalism. Within the ideology there is clear support for "good" institutional racism.
Would MLK have approved of that or would he have sought to abolish all forms of institutional racism fearing allowing any racism is ultimately to invite oppression.
Would MLK support mandated diversity targets based on skin color, gender, etc. or would doing so be another example of "good" institutional racism.
I think Malcolm X is more aligned with modern progressive liberalism. His belief in a society where power was distributed not based on talent, but somehow proportionately, fits much closely with progressive ideology and its embrace of identity politics. In essence, Malcolm X seemed to sanction the forced allocation of power and income to achieve an equal outcome "by whatever means necessary".
In a surprising way, Malcolm X and MLK are at the center of the two ideologies that have polarized this country today. One striving to eliminate all forms of discrimination to create a world where every individual citizen has the same citizenship rights. While the other ideology striving to produce an equitable outcome defined as an equal distribution of wealth and power across all recognized identity groups.
To understand the difference...
Imagine 1000 applicants for 20 slots in a medical school program. The applicants are an incredibly diverse group in every sense of the word. You can only accept 20 applicants. Based on the objective criteria that history has established as predictive of success, you sort the individuals 1 through 1000. All applicant demographic information has been scrubbed from their applications and the result of the ranking is not reflective of the society from which the applicants are drawn.
How would MLK vs Malcolm X have determined the 20 applicants that should be accepted?