At the beginning of “Stephen Sondheim’s Old Friends,” the Broadway stars who headline this new revue celebrating one of musical theater’s greatest giants, Bernadette Peters and Lea Salonga, address the audience to explain the show’s agenda. “Steve was always a ‘Broadway Baby’ at heart,” Ms. Peters notes, referencing one of his song titles, “and wanted his shows to be enjoyed by as many people as possible.”

The subtext here is that Sondheim — who died in 2021, after a nearly seven-decade career in which he revolutionized the musical without ever losing sight of golden-age ideals — did not, as some of his critics maintain, write coldly cerebral songs and shows. This isn’t news to any of us who have cherished the warmth and beauty of Sondheim’s words and music, the sense of wonder and yearning they convey: qualities even more central to their enduring resonance than his legendary wit.

Yet for Cameron Mackintosh, the British theater titan who began devising “Old Friends” before Sondheim’s death and is its producer — Ms. Salonga is tasked with plugging him in her opening remarks — mass appeal is clearly an abiding concern. While Mr. Mackintosh produced two previous tributes to the composer and lyricist, “Side by Side by Sondheim” and “Putting it Together,” he is better known for giving the world bloated mega-musicals such as “Les Misérables,” “Miss Saigon,” and Andrew Lloyd Webber’s “Cats” and “The Phantom of the Opera.”

For “Old Friends,” which arrives at New York after engagements on the West End and at Los Angeles, Mr. Mackintosh has recruited Matthew Bourne, a director and choreographer who shares his fondness for spectacle, to guide the actors and provide “musical staging.” (The minimal choreography is assigned to Stephen Mear; Julia McKenzie, a wonderful singer and actress who has appeared in numerous Sondheim musicals and directed and co-devised “Together,” is credited as an “artistic consultant.”)

Initially, “Old Friends” seems in danger of being hampered by the penchant for overstatement of its producer and director, and their obvious aim to convince skeptics that Sondheim was a people’s genius. Culling much of its material from “Side by Side” and “Together,” the new homage draws from musicals dating back to “West Side Story” and “Gypsy” (Sondheim penned lyrics for both while still in his 20s), often grouping tunes together to nod to specific shows, with some detours into less widely remembered projects.

“Company” is the first musical acknowledged at length: Beth Leavel and Gavin Lee, both accomplished musical theater veterans, dutifully mug their way through “The Little Things You Do Together,” a bitingly hilarious homage to marriage; it’s as if Mr. Bourne instructed them to overplay each lyric and gesture so that matinee crowds would get Sondheim’s jokes.

A too-cute “You Could Drive A Person Crazy” is followed by “Live Alone and Like It,” featured in the 1990 film “Dick Tracy,” and “Loving You,” an exquisite ballad from Sondheim’s 1994 musical “Passion,” sung nicely enough by Ms. Salonga but nearly sabotaged by an overly fussy arrangement by Stephen Metcalfe.

“Into the Woods” and “A Little Night Music” are referenced next, but it was a segment devoted to “Sweeney Todd” that first made me smile. Jeremy Secomb, a mighty-voiced actor who has played the avenging barber more than once, is cast opposite Ms. Salonga, whose own singing, darker and brassier than it was in her youth, is perfectly suited to Sweeney’s eager accomplice, Mrs. Lovett; the actress also brings a deft comedic facility to their numbers.

Ms. Peters, who has been closely associated with Sondheim for decades and has introduced roles in his shows on and off-Broadway, is similarly winning in lighter moments, particularly when she channels a jaded stripper in “You Gotta Get a Gimmick,” from “Gypsy.” But for me, at least, this widely adored performer’s abilities as a song interpreter have seldom matched her overall winsome charm; listening to her carefully deliver “Send in the Clowns” and the heartrending “Losing My Mind” — both of which she’s sung in Broadway revivals, of “Night Music” and “Follies,” respectively — I couldn’t help but wonder if any of the less celebrated cast members might have given them more punch. 

Mr. Bourne has, to his credit, recruited such fresh talent for this production. Several younger players — among them Jacob Dickey, Kyle Selig, and Maria Wirries, whose sterling soprano leaps out repeatedly — contribute to a soaring rendition of the “Tonight Quintet” from “West Side Story,” and the whole company harmonizes gorgeously during ensemble numbers such as “Sunday,” from “Sunday in the Park with George,” and “Being Alive,” from “Company.”

We also, briefly and at the very end, hear Sondheim singing, and we see him in a couple of photo and video montages, with projection design provided by George Reeve. One collection, projected as the cast performs “Not a Day Goes By,” from “Merrily We Roll Along,” shows the revered artist as a little boy and follows him into young adulthood and later in life. 

For anyone who knows something about the personal pain that informed Sondheim’s work, this is deeply poignant. His songs can capture emotion with just as much purity and directness, and I wish Messrs. Mackintosh and Bourne had trusted in that a bit more.