Musical theater giants from Leonard Bernstein, Betty Comden, and Adolph Green to John Kander and Fred Ebb have been drawn to “The Skin of Our Teeth,” Thornton Wilder’s Pulitzer Prize-winning opus. It’s easy to understand both their attraction to the fantastical, allegorical play, originally staged in 1942, and why, up until now, no one has succeeded in getting a musical adaptation produced.

Last presented on Broadway three years ago, for the first time since a very short-lived revival in 1975, “Teeth” covers enormous ground both tonally and historically, taking audiences from the Great Ice Age to an evocation of the biblical floods confronted by Noah to a more recent, man-made catastrophe, clearly based on the world war that was raging when Wilder wrote the three-act piece.

While the play’s focus is on one family — Mr. and Mrs. Antrobus, who have been married for thousands of years, and their children — other characters include Moses and Homer, as well as a dinosaur and a mammoth, posing all manner of possibilities for kooky song-and-dance numbers, not to mention fun costumes. And the fourth wall is repeatedly trampled, which surely enticed Kander and Ebb, given their debt to Bertolt Brecht and Kurt Weill.

I suspect that Ethan Lipton, a playwright, songwriter, and performer who has been a downtown favorite for years, also had Brecht and Weill in mind when he set about crafting the music, lyrics, and book for “The Seat of Our Pants,” the adaptation-cum-parody of “Skin” now playing at the Public Theater. Even Daniel Kluger’s orchestrations for Mr. Lipton’s folk-based, jazz-tinged songs, some of which are quite fetching, offer darkly brassy flourishes and a purposeful lack of sentimentality that bring Weill’s tunes and shows to mind.

But the epic theater delivered in “Seat” can be of a distinctly and unfortunately modern variety: a little smug, soaked in irony, winking at the audience as if to say, “You’re smart enough to get all this, right?” An announcer played by Andy Grotelueschen as a sort of aging hippie sings and delivers wisecracks at the beginning of each act, two of which are presented after a single intermission.

The production’s many nods to Wilder’s meta-theatricality include inside references to the careers of cast members, among them beloved and rising musical theater stars such as Ruthie Ann Miles, Shuler Hensley, and Micaela Diamond. During Act Two, the announcer assumes the role of playwright and delivers a monologue ostensibly explaining Mr. Lipton’s approach to the source material.

“I’ve found that many of its less appealing features,” the announcer opines, obviously referring to “Skin,” “are not essential to its character, and can be reconsidered or opened up or taken to the dump — without sacrificing any of the original’s bats— grandeur. … And my hope is that if we can make this glorious structure habitable enough for us to live in today, then it can show us ourselves today.” There’s plainly some cheek here, but not enough to sap the pretension poking through those italics. 

Director Leigh Silverman, who has worked frequently with both Mr. Lipton and the Public, has nonetheless culled impressive performances from her players. Mr. Hensley, who has been hilarious and moving as heavies ranging from Jud in “Oklahoma!” to the Grinch, brings both amusing haplessness and an edge of menace to Mr. Antrobus, a chameleon who takes credit for inventing the alphabet and the wheel but often seems out of his element, to put it delicately, as a husband and father. 

Ms. Miles lends her usual elegance to the part of Mrs. Antrobus, who is chafing after years as a devoted and much put-upon wife and mother; her son was originally named Cain, and changing it to Henry hasn’t done much good. Her maid, Sabina, has managed her own suffering with more sass, which Ms. Diamond channels in a sly, witty portrait.

As she did in “Skin,” Sabina adopts another guise and addresses the audience more frequently than her employers. “This show is still being written,” she insists as the musical draws to a close, before joining the rest of the company in a cleverly staged number that ends the production on a high note.

It’s a reference to the show’s central theme, also that of Wilder’s play: that for all the brutality we’ve unleashed and endured, humanity has demonstrated tremendous resilience. But I’d like to think that Mr. Lipton would consider tweaking “The Seat of Our Pants” — trimming the post-intermission portion, for starters, and then inviting Ms. Silverman and her talented actors back to the playground.