NBC’s holiday-season live-musical franchise is so proudly square that it’s surprising it took this long for them to get to “Annie.”

The recent tradition, launched by the producers Neil Meron and the late Craig Zadan, of ceding a night of network primetime to a scrappy, let’s-put-on-a-show broadcast seems to have found its perfect match in “Annie.” The family-oriented musical has never had the remotest claim on coolness. And, coming as it did in a flickering moment in which a production of this sort felt possible but still novel, “Annie Live!,” starring Celina Smith as its optimistic orphan, made its goofy eagerness to be liked, and its occasional raggedness, into assets.

The show’s staging allowed songcraft and showmanship to be the stars; numbers played out against minimalist backdrops, and the live audience was largely heard in eruptive applause but seen only in shadow. The scant amount of stage dressing — an American-flag scrim for Annie’s meeting with Franklin Roosevelt, a bouquet on an end table and a staircase to evoke the grandeur of Daddy Warbucks’ home — seemed intended to rhyme with the show’s message of scrappy resilience and high-spirited hope. Even as viewers surely understood they were watching a broadcast bolstered by the resources of NBC, it was possible to believe that this was something like the best sort of community theater.

Which means that certain flaws could get written off by the viewer somewhat easily. Before moving on to the full-throated praise, it’s worth noting that, say, Harry Connick Jr.’s iffy styling in a surrealistic bald cap pulled focus, and was best explained away as an occupational hazard of trying to make a star ready to play the role of a famous cueball. Connick’s occasionally being a beat behind on lines suggested, to the charitably inclined viewer, just how far the cast in general had come in order to bring this production to air at all.

It feels unsporting, after all, to pick on a production that seemed so resolutely determined to entertain — and one whose key flaws may be inherent in the source material, a show one loves, if one does, because of its flaws as much as despite them. “Annie” is somewhat front-loaded, with a draggy middle, but cast members including a polished Nicole Scherzinger as Warbucks’ secretary Grace and a lightheartedly wicked Tituss Burgess as greedy Rooster took the opportunity to push through it in their respective ways. Taraji P. Henson’s performance as an unusually vituperative Miss Hannigan — alternately both brutally nasty and deliriously pleasure-seeking even relative to past interpretations of the character — seemed like an actor taking the bet presented by an overwritten character and doubling down.

Which made young Smith’s performance the most intriguing of contrasts. While she was certainly polished, hitting her dance cues with aplomb and in strong voice, Smith was, for lack of a better term, appropriately kid-like. She brought to “Annie Live!” sweetness and a certain naivete, an innocence to the machinations of adults around her that seems essential to the part.

And Annie’s being recast once again as a young Black girl — after the 2014 film adaptation featuring Quvenzhané Wallis as Annie and Jamie Foxx as her newfound father figure — suggests a broadening in the number of children who can see themselves at the center of the story, singing and dancing and looking forward to tomorrow. This “Annie” treats its Annie being Black (and her savior Daddy figure being white) as matter-of-fact. This is so inherent in the story that it needn’t be dealt with further. And viewers were likely charmed into letting “Annie Live!,” so effectively committed to spreading joy, leave things there.

And why not? After all, this “Annie’s” arrival seems particularly apt — not just because audiences may have grown somewhat impatient for the precise sort of thing this NBC franchise offers. (This is the first full-scale live musical to be broadcast during the holidays by NBC since the similarly upbeat “Hairspray” in late 2016.) “Annie,” in its first Broadway iteration, ran nearly six years, cutting a sunny swath through the so-called national malaise of the late 1970s. And the story it tells of virtue and good-heartedness easily rewarded despite the apparent direness of circumstances is one to which the culture keeps returning, and one that might feel especially apt right now. Plenty of people listening to “Tomorrow” have reason to regard the past two years or so of yesterdays as especially unworthy of being commemorated in song. That made “Annie Live!” seem like the beneficiary of the best thing a work of theater can have on its side: Good timing.