In the final month of his 2020 presidential campaign, Joe Biden stood before a drive-in crowd in Toledo, Ohio, and announced he had “a chip on my shoulder” about people with fancy college degrees.

He would, Mr. Biden said, be the first president in “80 or 90 years” without an Ivy League degree — an exaggerated biographical detail that spoke to the image he sought to convey as the blue-collar, workingman’s candidate.

“I went to the University of Delaware, I was proud of it,” Mr. Biden said. “Hard to get there, hard to get through in terms of money. But folks, since when can someone who went to a state university not be qualified to be president?”

Mr. Biden — the first president without an Ivy League degree since Ronald Reagan, a Eureka College alumnus who left the White House 32 years before Mr. Biden entered it — has now set his administration on a collision course with Harvard, one of the Ivy League’s flagship universities.

His administration’s fight, in the form of a civil rights investigation into Harvard’s legacy admissions process by the Education Department, gives Mr. Biden an opportunity to show himself opposed to the country’s elites as he ramps up a presidential campaign in which he will need support from working-class voters culturally far afield from the Ivy League.

The inquiry serves as an early bank shot for Mr. Biden to show voters that his administration is trying to do something to respond to the Supreme Court’s ruling last month gutting affirmative action in higher education — a decision that led Mr. Biden to declare, “This is not a normal court.” The department’s Office of Civil Rights has significant enforcement authority and Mr. Biden, should he choose to use it, has the White House bully pulpit to negotiate a settlement with Harvard.

This week, the Education Department is hosting a “national summit on equal opportunity” in Washington. Mr. Biden has asked the department to produce a report by September with proposals of what the government should do in response to the court’s decision and singled out legacy admissions as an issue of concern.

But while the Biden administration’s investigation into legacy admissions will surely grab attention among a political and media class overrepresented by Ivy League alumni, it is far less likely to address enduring roadblocks to higher education like skyrocketing tuition costs and mountains of debt incurred by students.

“It’s a Band-Aid,” said Melissa Byrne, a Democratic activist who has spent years campaigning to make public undergraduate schools free and for the federal government to waive student debt. “The hindrance to higher education isn’t legacy admission, it’s a paywall that makes education a gift from parents to children or a debt sentence.”

People involved in the campaign to make higher education more equitable and accessible described the question of legacy admissions as limited to a few applicants to elite universities. Even at Harvard, according to court documents, legacies make up less than 5 percent of applicants, though about 30 percent of them are admitted each year — an acceptance rate more than seven times as high as that of the general applicant pool.

At less competitive schools, often state universities, legacy students are recruited and celebrated. Mr. Biden’s undergraduate alma mater, the University of Delaware, offers scholarships of up to $2,000 to children of the school’s alumni to help “enrich their education.” In 2018, a university publication wrote glowingly about an incoming student related to the school’s “legendary football coach” who said “it feels like U.D. is in my blood.”

Liz King, the senior program director for education at the Leadership Conference on Civil and Human Rights, said the Education Department’s civil rights office had been obligated to begin an inquiry about Harvard’s legacy admissions process after receiving a complaint about it. She said she hoped the Biden administration would not limit its higher education investigation to legacy admissions, but instead look broadly at a system she described as discriminatory for students and applicants of color.

“We cannot look at the binary question of a legacy check box as an opportunity to wash our hands of the negative, the real threat posed by the Supreme Court decision,” said Ms. King, whose alma mater, Wesleyan University, ended legacy preferences in admissions last week. “What we need is equal access in higher education.”

The country’s elite universities will soon face growing public pressure to reveal more about what have long been an opaque admissions practices, said Art Coleman, who led the Education Department’s civil rights office during the Clinton administration and now serves as a consultant to universities and other nonprofit organizations.

“In this universe of total distrust of systems, including institutions of higher education, there is a notion of this mysterious black box that no one understands and few people trust,” Mr. Coleman said. “My hope is we’ll seize the moment now and it will lead to better transparency around well-developed practices.”

Going after elite schools’ legacy admission policies would seem to be a bipartisan political winner.

After the Supreme Court’s decision, Senator Tim Scott of South Carolina, a Republican running for president, called for Harvard to end its legacy admission policies. On Wednesday, Senator Jeff Merkley of Oregon and Representative Jamaal Bowman of New York, both Democrats, introduced legislation to stop universities from giving preferential treatment to children of alumni and donors. (Their bill, however, did not have any Republican co-sponsors.)

Last year in Connecticut, Gary Turco, a Democratic state representative, introduced legislation to prohibit legacy admissions at the state’s universities. At a public hearing, Yale’s undergraduate admissions director delivered a defense of its policies, but Mr. Turco said that besides officials from the state’s private universities, the only feedback he had heard was supportive.

“I hear that often from people that they want a fair, level playing field where everyone has the same opportunity from the start,” Mr. Turco said. “They want it based on an individual’s merit and hard work and based on somebody’s need versus these legacy traditions, which have created a lot of the social economic inequities that we see in so many areas of society.”

The idea of legacy admissions is also far easier to comprehend than other more insidious obstructions to higher education. Standardized tests, substandard high schools and wealth inequality all serve as more significant factors in education equity efforts, said David Hinojosa, the director of the Educational Opportunities Project at the Lawyers’ Committee for Civil Rights Under Law.

“From the public’s perspective, they can grasp more easily the inherent inequities with legacy admissions and how those are entrenched with privilege but no merit,” Mr. Hinojosa said. “People don’t understand the history of standardized tests.”