To follow up on Matt and Ezra, another angle to run at it from is to apply the logic to the civil rights movement. If one takes the Rauch/Brownstein argument seriously, wasn’t it also wrong to “federalize” the divisive issue of segregation? Of course, they wouldn’t say that, but that first of all demonstrates the underlying question-begging; as a normative matter, 99% of the time saying something should be “left to the states” is just another way of saying that the question of social justice isn’t a very high priority. But even more importantly, it also raises obvious problem for their empirical claims: desegregation and disenfranchisement were “protest issues” even when they were left to the states, and became “ordinary politics” issues after they were federalized. And to get on my old hobbyhorse, the gentility of the American abortion debate pre-Roe has been grossly romanticized, and it’s also worth noting that Canada has federalized both abortion and gay marriage, and not only have the issues remained largely “ordinary politics” they aren’t even especially salient, and at least with the former the outcome has been perfectly stable. This suggests that federalism isn’t the key variable here. There’s no reason to believe that allowing 20 states to ban abortion will somehow diminish the conflict over abortion, and of course you have the negative externality of many women being maimed or killed in black market abortions, arbitrarily forced to carry pregnancies to term, etc.
And, of course, there’s the larger issue: why “protest politics” is supposed to be a problem in the first place. People protesting and mobilizing around what they consider to be fundamental injustices, at least in the context of a nation where the basic legitimacy of the state isn’t in question, is the sign of a healthy polity, not a dysfunctional one.