Still, he acknowledged that Republicans had “figured out how to message this.”

The messaging goes back to Mr. Trump, who, in the final weeks of the 2020 campaign, announced the formation of the 1776 Commission, set up explicitly to link what he said was “left-wing indoctrination” in schools to the sometimes violent protests over police killings.

A report by the commission was derided by mainstream historians; Mr. Biden canceled the project on his first day in office, but its impact endures on the right.

Media Matters for America, a liberal group, documented a surge of negative coverage of critical race theory by Fox News beginning in mid-2020 and spiking in April, with 235 mentions. And the Pew Research Center found last year that Americans were deeply divided over their perceptions of racial discrimination. Over 60 percent of conservatives said it was a bigger problem that people see discrimination where it does not exist, rather than ignoring discrimination that really does exist. Only 9 percent of liberals agreed.

Some Democratic strategists said the issue was a political liability for their party. Ruy Teixeira, a senior fellow at the left-leaning Center for American Progress, recently wrote, “The steady march of ‘anti-racist’ ideology” into school curriculums “will generate a backlash among normie parents.”

In an interview, he criticized leading Democrats for not calling out critical race theory because of their fear that “it will bring down the wrath of the woker elements of the party.”

In Loudoun County, Va., dueling parent groups are squaring off, one that calls itself “anti-racist” and the other opposed to what it sees as the creep of critical race theory in the school district, which enrolls 81,000 students from a rapidly diversifying region outside Washington.

After a 2019 report found a racial achievement gap, disproportionate discipline meted out to Black and Hispanic students, and the common use of racial slurs in schools, administrators adopted a “plan to combat systemic racism.” It calls for mandatory teacher training in “systemic oppression and implicit bias.”