In hindsight and on paper, it actually makes a lot of sense: The Replacements were a notoriously volatile and sometimes downright magical live band, so why not put out a live album to help sell them to the masses?

Of course, most of the sensible ideas in the pothole-riddled career of Minneapolis’ famously underappreciated quartet never quite worked out the right way. And so here we are, 31 years later, finally getting our hands on and ears damaged by “For Sale: Live at Maxwell’s 1986,” the first true live collection in the four-decade history of the ’Mats.

“Everybody is asking me now, ‘Why didn’t anything come of this?’ ” said Michael Hill, the former Sire/Warner Bros. Records executive who oversaw the original recording. “And as I listen back to it now, I’m asking the same thing.”

The 29-song, 83-minute collection — out next Friday — is taken from a single gig at a low-rent but much loved New Jersey club, four months after the release of the “Tim” album and just a few more months before co-founding guitarist Bob Stinson was ousted from the group.

Unlike most of the widespread concert bootlegs from the Replacements’ original 1980-91 run, this one was captured with a hi-fi mobile recording unit. The audio quality is surprisingly terrific, especially after some new remastering.

Perhaps even more of a shock, the Maxwell’s performance itself was superb, with very little of the knucklehead antics and half-finished songs that became a trademark of too many Replacements live shows at the time.

While he has no stake in the band or the record company these days, Hill was deeply invested in trying to make stars of the Replacements in the late ’80s as their artist and repertoire representative at Sire. He was among the handlers who — in the aftermath of the band’s now legendary 1986 “Saturday Night Live” performance (which got them banned from the show) — pinned down a plan to record them just a few weeks later.

They chose Maxwell’s in Hoboken, Hill said, because “it had the vibe of the CC Club and other low-frills places you’d be prone to see them at back in Minneapolis, but it also sounded great in there.” (And trying to record the band onstage in Minneapolis undoubtedly would’ve had messy results, as anyone with bootleg copies of First Ave shows will tell you.)

Hoboken area photographer and music writer Caryn Rose remembered the Maxwell’s show being announced relatively last-minute, which added to the buzz around the gig. The truck parked out front was the only clue to attendees that the show would be recorded.

“They were playing bigger venues by then, so it actually felt important despite it being at Maxwell’s,” said Rose, whose photos feature in the new two-CD/LP set from Warners’ collectors label Rhino Records. “It wound up being a room full of mostly die-hard Replacements fans.”

That much is obvious listening to the recordings today. Fans can be heard rabidly yelling out requests such as “Sixteen Blue” (never played) and “Color Me Impressed” (already played when some clown yells it out 18 songs later in the recording). The in-the-know crowd even calls for cover songs the band was known to play, immediately obliging a woman who wanted Sweet’s “Fox on the Run.”

Being one of those die-hards, Rose said she had grown tired of “talking up the band to somebody who’d then turn out to see them, and the show would be a total mess.” This wasn’t one of those disappointing performances, though, despite nervousness that the band might intentionally muck it up.

From the opening tune “Hayday” — in which frontman Paul Westerberg repeatedly howls “Murder!” (an in-joke akin to a cavalry charge call) — there’s a full-tilt, nonstop energy to the set. It’s an absolutely thrilling document of the band, the only disappointment being that it sat on the shelf for so long.

The blistering, full-throated versions of “Dose of Thunder” and “Hold My Life” near the start of the album arguably surpass the studio versions featured on “Tim.” Most of the songs played off the prior record, 1984’s “Let It Be,” come off with a similar thunder, including “Favorite Thing,” “Tommy Gets His Tonsils Out” and especially “Gary’s Got a Boner.”

Arguably the best — or at least the leanest and meanest — tracks on “Live at Maxwell’s” are the punkier, brasher songs from the quartet’s first two releases for Minneapolis-based Twin/Tone Records, including “Johnny’s Gonna Die,” “Otto,” “Goddamn Job,” “Go” and “I’m in Trouble.” In fact, the primo quality of that early material may have been a detriment in Sire Records’ eyes.

“The songs from the Twin/Tone albums sound superior, and obviously that wasn’t really a selling point to the band’s new label,” Hill said. Mostly, though, he said, there’s really no concrete explanation why the Maxwell’s recording was never released. “It was never a sure thing, and got more unsure as time went on,” he said, citing a “general uncertainty” at Sire over how to market it and then “the unforeseen internal strife within the band.”

Within months, the band — then still anchored by drummer Chris Mars and bassist Tommy Stinson — parted ways with longtime manager Peter Jesperson. Then it kicked out Bob Stinson (Tommy’s older brother) as the guitarist’s alcohol and drug abuse and mental illness supposedly became too problematic even for a band rife with those ailments.

“Live at Maxwell’s,” however, offers solid proof Stinson could still shred like a NASCAR tire and maintain a delightfully unhinged groove near the end of his tenure with the ’Mats — or at least he could on a good night like this.

“It really highlights how Bob and Paul played off each other, with each other and against each other,” said Bob Mehr, author of last year’s bestseller biography “Trouble Boys: The True Story of the Replacements,” who also helped produce the live album and wrote the liner notes.

Well versed on ’Mats live bootlegs, Mehr rated this Maxwell’s set as “head and shoulders above all the other recordings in terms of fidelity.” Sire did fund another hi-fi live recording five years later during the “Don’t Tell a Soul” tour, resulting in a very limited-edition and also quite terrific promotional EP titled “Inconcerated.”

While also fond of “Inconcerated,” Hill believes that in the end, it’s weirdly fitting that “Live at Maxwell’s” came out in 2017 instead of 1987.

“The original ’Mats fans — God love ’em — are mostly still with us, and now there’s a new generation of fans,” Hill said, pointing to the hoopla around Westerberg’s and Tommy Stinson’s 2014-15 reunion run as the Replacements. He also pointed to changes in the music industry that may benefit the release.

“A lot of the roadblocks that got in the way of the Replacements being a successful, independent band in the ’80s don’t exist anymore,” Hill said, referring to the raw, rough-and-tumble approach heard throughout the new collection; traits you never would’ve heard on the almighty radio in the late ’80s.

“The qualities that the label didn’t find useful back then are exactly the things that make this recording unique and special today.”