The story behind Stephen Sondheim and George Furth’s “Merrily We Roll Along” is nearly as poignant, and certainly as renowned in theater circles, as the show’s plot. 

Sondheim and the director and producer Hal Prince had just spent a decade, the 1970s, revolutionizing musical theater with triumphs ranging from “Company” to “Sweeney Todd,” when they began working with librettist Furth on an adaptation of Kaufman and Hart’s 1934 play following a successful playwright backwards in time, from jaded middle age to starry-eyed youth.   

“Merrily” the musical opened in November 1981 to largely negative reviews. The problem, many agreed, was less Sondheim’s score—which introduced such enduring gems as “Good Thing Going,” “Old Friends,” and “Not A Day Goes By”—than Furth’s book, which focused on three friends pursuing literary and show business careers. 

“We keep waiting for some insight into these people,” wrote then-New York Times critic and longtime Sondheim champion Frank Rich, “but all we get is fatuous attitudinizing about how ambition, success, and money always lead to rack and ruin.”

The production closed after a mere sixteen post-preview performances, and Sondheim and Prince would not work together again for two decades. Yet while the show has never been revived on Broadway, directors have, over the past 40 years, been drawn to it like flies to honey, each determined to overcome its challenges. 

Maria Friedman, the veteran British actress now helming a starry off-Broadway staging of “Merrily,” comes to the task with more experience than most: Ms. Friedman starred in a 1992 production featuring revisions by Sondheim and Furth, and made her professional directorial debut twenty years later with a well-received revival retaining those changes, at London’s highly regarded Menier Chocolate Factory.

For the new “Merrily,” the script, with a few small tweaks, has been passed to Daniel Radcliffe, Jonathan Groff, and Tony Award winner Lindsay Mendez, along with a troupe of supple supporting actors and singers whose names are less well known. Ms. Friedman’s deep affection for them, and for the material, is palpable.

The show opens in the late ‘70s, as Mr. Groff’s character, Franklin Shepard, a once-respected composer now intent on producing vapid Hollywood movies, is throwing a party to celebrate the release of one such flick. His old buddy Mary Flynn, a once-rising novelist, is there practicing her current vocation, alcoholism. “You are all junk,” she tells her fellow revelers, after having several too many. “And you,” she adds, facing Frank, “you deserve them.”

Mr. Radcliffe’s Charley Kringas — a playwright and Frank’s former songwriting partner, and the third member of their once-inseparable trio — hasn’t bothered to show up at all. As we will learn in the following scene, set a few years earlier, he and Frank haven’t spoken since Charley berated him — during a live television interview — for caring more about commercial prospects than their creative or personal relationship.

The musical’s reverse journey ends with Frank and Charley first meeting Mary, just as the very young men decide to become collaborators. All three are full of wonder and hope, for themselves and for the world they’re inheriting.

I didn’t catch the original “Merrily,” but I’ve left most of the various productions I’ve seen more moved by this heartbreaking arc, and by the melodies and lyrics informing it, than bothered by the glitches that many critics have emphasized. 

And Ms. Friedman’s is both as piercing as any I’ve attended and refreshingly playful, stressing the characters’ quirks and foibles in ways that makes their progress — and lack of it — highly accessible.

Indeed, whatever star power the three leads bring to this staging, it never overshadows the authenticity of their performances. Mr. Groff might have it easier in this sense than the others, as Frank is meant to be intensely charismatic and something of a celebrity; but the actor shows us the regret and seemingly bottomless need underlying his selfish arrogance.

Mr. Radcliffe, who proved he had musical-comedy chops in a 2011 revival of “How to Succeed in Business Without Really Trying,” vividly charts Charley’s evolution from sweet young nerd into sardonic veteran, tearing his way through Sondheim’s acid-soaked “Franklin Shepard, Inc.” 

Ms. Mendez may be the most affecting, though; her effortless transparency makes Mary’s sense of dejection—the product of her unrequited love for Frank—as clear as her scathing wit.  

Under Ms. Friedman’s stewardship, the stars also convey the abiding rapport these three share, and the sheer fun they can have in each other’s company. 

Tim Jackson’s loosey-goosey choreography and the jazzy orchestrations of Sondheim’s longtime colleague Jonathan Tunick (presented here under the music direction of Alvin Hough, Jr. and supervision of Catherine Jayes) also provide buoyant relief.

After all, as we’re reminded in “Now You Know,” the ironically upbeat number that closes “Merrily”’s first act, “Life’s a killer,” but you can grow by surviving its punches. And if you don’t, you can at least try to enjoy the ride.