It takes nerve to make a documentary about the most unpopular period of a massively popular public figure’s life. “One to One: John & Yoko” demonstrates that it’s worth the effort.

Co-directors Kevin Macdonald (“The Last King of Scotland”) and Sam Rice-Edwards have done an impressively deep archival dive to give us this portrait of John Lennon in 1972, the year the ex-Beatle arrived in New York City to stay and embarked on a period of radical politics and art. It was an era of upheaval in American society and a time when Lennon tried to leverage his celebrity to effect change, in ways that made him look simultaneously sincere, committed and naive.

A chief reason for his unpopularity at this time is right there in the film’s title: Yoko Ono was with Lennon every step of the way as wife, muse, collaborator and co-instigator, and her presence drove most people nuts. Already assigned blame (wrongly) for the breakup of the Fab Four, Ono pushed buttons simply by existing, as a woman, as an avant-garde artist, as a Japanese “witch” (about the nicest thing she was called) who’d somehow hypnotized the Smart Beatle into becoming a slave to the radical left.

By contrast, what do we see in the footage gathered for “One to One”? A couple deeply in love, isolated by fame and equal partners in trying to figure a way out of the world’s bag of expectations. Sometimes the bag was literal, as in art events that had the couple climbing out of a large black sack. (There’s footage of one “Bagism” performance in the film, and at this distance, it’s more weirdly moving than silly.) Often it was rhetorical: Lennon and Ono made a point of going on daytime talk shows, engaging hosts like Mike Douglas and Dick Cavett in political discussions and trying to raise Middle America’s consciousness while showing that they were just another normal couple trying to get by.

The times, of course, were anything but normal. As edited by Rice-Edwards and Bruna Manfredi, “One to One: John & Yoko” is as much a high-impact history lesson as a Beatles doc, with the events of late 1971 and 1972 piling on, one after another, starting with the Attica Prison riot and its horrific aftermath. It was an election year; the war in Vietnam was grinding on; and in the wake of the 1970 Kent State killings, the protest movement was simultaneously splintering and turning more radical. Richard M. Nixon won reelection even as whispers about the Watergate break-in grew louder.

The documentary gives us snippets of all these developments, as well as quick blasts of TV shows (“Let’s Make a Deal,” “The Waltons”), advertisements (Frosted Flakes, Tupperware), visitations by George Wallace and Shirley Chisholm — it’s a nostalgia tour with the knives out and/or a history lesson to teach the kids that America’s always been divided and that the good old days were never.

Within the film’s diorama, John and Yoko come off as both brave and deluded. It didn’t help that Lennon allied himself with some of the most doltish cultural and political voices on the left, including Jerry Rubin of the Yippies; Dylan-obsessed “garbologist” A.J. Weberman; and David Peel, Weberman’s co-founder of the “Rock Liberation Front” and the most embarrassing ad for pothead folk musicians ever.

To his credit, Lennon was pulling away from this group by mid-1972, disenchanted with their plans to riot at the Republican National Convention in Miami. (“You want to overthrow WHAT and replace it with WHAT?” he’s quoted as saying.) By then, the U.S. government was moving to deport the couple, and Lennon and Ono had turned their attention to other causes, including two concerts at Madison Square Garden to benefit children with intellectual disabilities at the Willowbrook State School, where deplorable conditions had been recently exposed by Geraldo Rivera.

The film of those concerts — the last full-scale performances Lennon ever gave — is returned to again and again in “One to One,” and it’s a revelation that makes his one official album of the era (the strident, artless “Some Time in New York City”) look like a pose. Backed by Ono and a full band, Lennon tears through performances of “Instant Karma!,” “Cold Turkey” and other solo singles; hushes the crowd for a deeply emotive “Mother”; and plays “Imagine” as if the paint were still wet.

Ono has her moments as well, and 50 years on, they’re easier to contextualize and appreciate, especially the exhausted “Looking Over From My Hotel Window,” sung by a woman who has lost a young daughter to a vengeful ex-husband, had several recent miscarriages and is well aware that she’s one of the most reviled women in America. The filmmakers take pains to present her as one half of a couple living in public the best they know how, sometimes foolishly, at other times with remarkable forbearance.

By March 1973, Lennon and Ono would move uptown to the Dakota, and the next chapter would begin, but “One to One” retrieves this chapter from the cultural memory hole and restores it to its proper complicated place.