This is quite a good introduction to CRT--and it's an important question.
Critical race theory has some really important insights but should not be taught in schools, including graduate law (except as history) owing to its numerous enormous problems. Instead, the insights need to be incorporated as quickly as possible into a framework that allows knowledge to be built and refined.
The first order of business is to completely divorce critical race theory's insights from critical theory, which is in its original conception from Horkheimer and colleagues was an abject failure of pragmatic epistemology, and got increasingly worse as later scholars incorporated increasing amounts of postmodernism. Two aspects of critical theory stand out as particularly atrocious. First, critical theory embraces--practically worships--the intellectual insight from critical theorists (individual biased people) as superior to ordinary science (a collective effort with multiple aspects of intentional sociology that foster increased accuracy in conclusions and in estimates of certainty). Secondly, critical theory demands that the critical theorist be an activist, meaning especially emotionally invested in the outcome of their work. This is an absolute disaster for human psychology: scientists train for years to be impartial while researching things that do not directly impact them and still mostly are unable to do it, needing their colleagues to rein in their flights of fancy; critical theorists first maximize their bias, and then throw away any formal method for checking their correctness save their own cognizance of error. This is incredibly corrosive to the search for anything like objective truth--and adding postmodern ideas where truth is blithely discarded in favor of truth-claims that have no particular grounding in reproducible observation just sinks the effort yet further. It's a complete disaster, albeit one often seductively presented. Keep it far, far away from anything important! Racial justice is important.
The second order of business is to extract the valuable insights from critical-race-theory-minus-critical-theory and jettison the numerous harmful and flawed approaches that the theory has accrued to itself.
The valuable insights are the same kinds of valuable insights that people usually get when they go, "Oh, huh...you mean this is a whole system? Maybe we should think about the whole system?" Ecologists have been doing this since forever. Microbiologists forgot the lesson of looking at the whole system while studying individual genes and pathways and cellular subsystems...until they rediscovered it and dubbed it "systems biology". When you have phenomena occurring in a complex system it is a very good idea to look at the whole system from time to time. We can't really give much credit to CRT for this insight given how many times we've had this insight before, but we can give them credit for using this insight to illuminate a variety of problems from systematically poor medical treatment to systematic failure to extend job opportunities. That stuff's really important.
However, the naive idea of teaching CRT more broadly presupposes that broad awareness of these things is part of the solution. This is almost unfathomable given what we know of human psychology. We are amazing at overgeneralizing patterns--just attend to some cue and get some input and you're going to form some subconscious opinions about how things like that go. That is, you'll come up with stereotypes. So what do you suppose is likely if you train everyone to be hyper-aware of race all the time? Furthermore, if one knows one is the target of negative stereotypes, being aware of the stereotype is likely to provoke stereotype threat, cognitive distress in response to the (perceived) pressure. So the default expectation is that paying attention to the kinds of racial issues that CRT tries to highlight will (1) worsen racial prejudices and (2) traumatize the already vulnerable. Spectacularly bad idea unless you can do something amazing in contrast (and you'd better demonstrate how amazing this is).
Furthermore, CRT has even in its most academic format incredibly bad epistemological grounding, and the simplification of that into popular culture makes things dramatically worse.
The worst of the worst is over-reliance on narrative as adequate to counter careful systematic observation. This gives rise to glorification of anecdote as "lived experience" and a practically worshipful attitude towards "speaking my truth". Anecdotes are useful for opening your eyes to unexpected possibilities. For almost everything else they are exceedingly unreliable--as we know full well to expect given how strong are confirmation bias, self-interest, tribal affinity, and so forth. CRT's embrace of narrative is, at the most generous, incredibly lazy intellectually; at worst, a cynical play that seeks to exploit our cognitive affinity for story to use incomplete truth and/or falsehood to garner support for changes that are not warranted by fact.
Second worst is the glorification of tribal factionalization and destruction of the idea of common humanity by over-reliance on the idea of intersectionality. As with CRT as a whole, intersectionality contains a very important (though not terribly novel) insight: not everything combines linearly. Sometimes if you are X and Y it is quantitatively different from what you'd expect from X alone plus what you'd expect from Y alone. For instance, if you are black and female, you may find conditions wherein neither your blackness alone nor your femaleness alone nor the simple addition of the two can account for what you experience. Okay--got it. This is true, everyone should have always expected it is true, that we didn't all get this is tremendously embarrassing given that this is like the bread and butter of statistical analysis and has been for many decades, and--it really was valuable that Crenshaw and others highlighted this and brought it out. Unfortunately, what you absolutely totally do NOT want to do given people's incredibly powerful tribal instincts to assume the best of their group and dehumanize and assume the worst of any other group is to dice people up into piles of groups so they have cause to hate and other pretty much anyone else in society who isn't exactly like them. That intersectionality is a deep part of someone's identity is incredibly corrosive to society and, if you decide to get all hard-nosed about it and start measuring predictive power, also seems rather wrong.
No-one should really be that surprised that this has then given rise to a toxic dehumanizing stew of terms and phrases and ideas like fragility, privilege, black-on-black (crime/killing), etc. etc..
I could say a lot more but--CRT is a disaster as a set of ideas.
Oddly enough, though, I have a lot of respect for the seminal figures in the CRT movement. Their writings are measured and thoughtful; they stress unity and appeal frequently to evidence. The legal system in which they were trained demands such, and when used to operating under those constraints, adding a small dash of CRT is actually quite helpful. But if you actually look at the premises of CRT as usually defined, there's no reason to be so constrained. Indeed, throughout Critical Race Theory: An Introduction (3rd ed; Delgado and Stefancic), you find constant little hints of going completely out of whack being allowed and maybe even encouraged by CRT as defined ("no white member of society seems quite so innocent", p. 91)...and also occasional self-reflection recognizing that CRT is drifting into unhelpful territory (e.g. p. 107), "Another issue that some crits raise is that the movement has become excessively preoccupied with issues of identity as opposed to hard-nosed social analysis." Well--indeed, that's a valid concern. But none of the principles of CRT embrace hard-nosed social analysis, so what should one expect?
Racial justice, as I said, is important. But at this point, after providing us with some key insights into ongoing injustice, CRT pulls almost entirely in the wrong direction. It needs to be discarded and re-imagined as "normal science" (as crits like to characterize racism). Ditch blanket characterizations of "racism is normal". What does that even mean? Quantify! How prevalent, how strong of an effect, in what circumstances? Either quantify the strength of "interest convergence" (i.e. the idea that whites won't do anything about racism unless it somehow benefits them--never mind that this is in strong contrast to all the times many people of multiple races have been driven by moral outrage or humanist compassion to change things even to their own detriment) or forget about it--just remember that if you want a change you have to motivate it, and yes, it's harder for people to embrace a change that is going to negatively affect them. Duh. This is true of every change ever. That race is a social construction doesn't even need to be stated any longer--we've tested (no thanks to crits, incidentally) and after correction for various environmental factors, the residual explanatory power of race as typically defined for pretty much anything people have cared to look at (IQ, income, whatever) is negligibly small compared to the person-to-person variability within groups. And so on and so forth.
Then, after we revamp CRT into something else (maybe "race systems analysis") to ditch its weird epistemology, societally destructive tendencies, aversion to quantification of effect sizes, embrace of methods easily dominated by the charismatic and subject to immense unquantified noise, and so on--after we turn it into something by construction much closer to what the founders of CRT did in practice, except made robust by quantification and study--then we can evaluate whether it's worth teaching in school. Preferably after it's proved itself by enabling us to help find solutions to racial justice problems--cause this stuff's hard, so maybe we'll mess it up again.
In the meantime, it's a complex and mostly wrong set of principles that have led to a mixture of valuable insights and destructive societal trends, and isn't suitable for teaching. Through a combination of poor quality control, overreach, and unexpected embrace and extension by society, CRT has also contributed to many of the strongest regressive tendencies in racial justice in the United States (both because of evoking backlash--some ill-considered, some fair--and because of on its own stoking animosity between former allies in the effort to increase justice). Ironically, for something prompted in large part by frustration with the slow pace of the classical liberal approach to racial justice actually delivering, it seems to me to have been worse than nothing despite a few important legal victories that did establish important principles.
We should, however, teach a realistic version of history--neither sugar-coated nor defeatest, neither lionizing everything nor demonizing it. But that's not really CRT. That's history. In order to counter the historical reality of people too often falling into tribalism, it's fair to emphasize the principle of common humanity as our ideal for now. Otherwise, history should be about what happened, roughly in proportion to how big an impact things had on what we find important now and/or is important to not repeat (plus what is needed to understand context), and since racial injustice is still a big problem (and always threatens to get worse--tribalism again!) there should be a decent amount of focus put on that.