1. Critiques do not make a philosophy, and the architects of postmodernism as is were not of the right as an as of now verifiable fact. That's like saying "Well, Nietzsche agreed with a few critiques of Hobbes, Hobbes is more to the right, therefore Nietzsche on the right."
This misses the point. Derrida was a man of the left, but what is called left-wing cultural critiques are just borrowed from the right and slightly repackaged.
2. We agree in principle, but there is a major problem there with the current strain of the left.
How can any culture or viewpoint evolve without criticism or objective testing from the... outside? The outside in any discipline offers the best view, but today's multiculturalism will not tolerate those criticisms.
It can't. And multiculturalism was a fad that has long-ago run its course on the left. It's mostly dead today.
"Black communities have X amount more violent crime than the national..." shouted down by a mantra of "it's poverty, history, and oppression!" no, those factors may contribute to the problem, but that is obvious false rationalization for the cultural factors that are leading to the violence.
Oh, that's silly. What you call "shouting down" just sounds like an expansion of the discussion and an attempt to answer a question. I would say that unnamed "cultural factors" are an obvious false rationalization designed to avoid confronting realistic but politically unpalatable solutions (though note that crime has been plunging around the country since the 1990s, meaning that we're on our way).
Adding to that, for the most part you're talking about a right from 1968. Most on the right want those subcultures to adapt if they want to have success, the left would demand that they keep their culture and that the right accept it carte blanche... or else be shamed back to their caves of intolerance.
Again, this is a really dated argument. As I mentioned, crime has been falling hard (seems like the decline has leveled recently). The big discussion among people who study it has been "why has it fallen so much, and why did it rise before?" rather than "oh what are we going to do about this crime problem?" And the more specific critique of your hypothetical right (that wants "those subcultures to adapt so they can have success") is that past policy still has a huge impact on economic success (for example, even after controlling for income and educational attainment, whites are still far wealthier on average, largely owing to much greater intergenerational wealth transfers--such as inheritance and parents paying for college/cars/down payments on homes, etc.). This is not rocket science. If you want to understand the issue, you'll be able to. If you want to shout down serious inquiry, you could miss it.
3. I disagree because we do, and I can readily demonstrate how it is cultural and not color based.
It doesn't matter if you disagree with my claim. You asked:
"If we look at something like race, what is the better way to maintain a sense of union? Is it to have a color blind society that does not judge on skin color, or to try to find supposed cognitive biases everywhere in an attempt to make a fair society?"
If you want to have a color-blind society that does not judge on skin color, you want to identify and address cognitive biases and make it so.
The right to it's credit is not interested in the color of your skin, they are interested in the culture they fought for and while not at all perfect in any respect, America has been a great force of freedom and civic greatness for generations, the left is unequivocally creating a complicated framework of who is privileged, shamed, oppressed, oppressor ECT...
That first sentence seems absurd if you've been following politics the past year. If you're a member of a big majority, you don't tend to define yourself in terms of it, granted. You're just a regular person. If you're not, you're constantly reminded of it. As America has become less white, more people have begun to think of themselves specifically as white. And that's where we get this flood of people dropping buzzwords like "white genocide" and "demographic replacement" that are totally alien to a liberal outlook, just as the left has moved away from that kind of illiberal outlook, outside of fringe elements on college campuses (that the far right gives greatly disproportionate attention to because they have a symbiotic relationship).
With little or no regards to character, it is all a redundant hierarchy of black/brown/yellow/green/pink, gay, and woman good, and old white, man, and straight intolerant and hateful until proven otherwise on the altar of New Left shame. That is not a fair system of thought and just as bad or worse than the dominant culture forcing conformity to its views for mainstream success.
You're stuck in the '60s, and missing what has gone on with the right (specifically, the fall of the religious right and the rise of the ethnonationalist "alt right" to a dominant place in Republican politics). The mainstream left is all about Enlightenment values. Even at the level of the average uninformed voter, admiration of science is extremely high (in the abstract). The climate change discussion really illustrates the gap. The left sees it as obviously true and hugely important, and the opposition as being irredeemably stupid. The right sees it as racial redistribution and sees the opposition as being evil.