About this time last year, Rick Hahn stood in front of the Chicago media corps and uttered the phrase that could come to define his tenure as White Sox general manager: “mired in mediocrity.”

Hahn had used the news conference to announce that the White Sox, a franchise stuck in neutral despite repeated bold attempts to contend, needed a significant change. He also gave baseball on the South Side a memorable slogan.

“Once I finished, our VP of communications said, ‘You had to use alliteration? You know that alliteration is going to be what’s taken away,” Hahn said in a recent interview. “But it was heartfelt. Although inadvertent, it was reflective of the conversations I had with others around the organization expressing similar frustrations with annually trying to patch this thing together, instead of trying to build something sustainable for an extended period of time.”

Since then, the White Sox have embarked on a scorched-earth rebuild that could set a new standard for how to repair a struggling franchise.

In December, they traded ace Chris Sale to the Boston Red Sox for a package that featured Yoan Moncada, perhaps the sport’s best prospect, and Michael Kopech, a pitcher who throws over 100 mph. A day later, they shipped outfielder Adam Eaton to the Washington Nationals for three more prospects, including former first-round draft pick Lucas Giolito.

Then in the past two weeks, they blew the rest up, swinging deals with the Chicago Cubs and New York Yankees that netted them seven more minor-leaguers in exchange for star lefty Jose Quintana, infielder Todd Frazier and relievers David Robertson and Tommy Kahnle. And there could still be more to come before the July 31 non-waiver trade deadline.

The White Sox, the American League’s worst team, have already acquired 15 prospects in trades over the last eight months, including three ranked in the top 12 in baseball by MLB.com. That, coupled with the signing of heralded outfielder Luis Robert, a 19-year-old from Cuba, has turned a unremarkable White Sox farm system into arguably the industry’s best—a stunningly rapid turnaround.

Cuban outfielder Luis Robert poses with a signed Chicago White Sox jersey. Photo: Charles Rex Arbogast/Associated Press

The promise of that future has started to arrive. The White Sox promoted Moncada last week to replace Frazier, with others likely to follow in the next year or two.

“We’re certainly pleased with how things have started off in this process,” Hahn said. “We know it’s going to take time, and we know we still have a lot of work to do.”

At this point, it’s impossible to deny that Chicago’s current approach—one that Hahn and executive vice president Ken Williams advocated for heavily—works.

The Cubs, the reigning World Series champions, and the Houston Astros, the AL’s best team, proved as much. Under the direction of progressive, data-driven leaders, both franchises opted to completely dismantle their rosters in order to rebuild from the ground up, enduring several seasons of misery to accumulate enormous amounts of talent.

Whether you call it “tanking” or simply a sound strategy, this much seems clear: The most successful rebuilds are the most aggressive rebuilds. Organizations that try to tread water drown; the ones that survive dive into the water headfirst.

After all, whether the White Sox lose 100 games or 85 games in a given season, the result is the same: They miss the playoffs. At least the 100-loss season leads to a coveted high draft pick, one of the few ways for teams to acquire potential stars before their primes. It’s that realization that has made these kinds of rebuilds more prevalent.

“When you look over the history of the draft, the majority of the impact players come from a certain area of it,” Arizona Diamondbacks general manager Mike Hazen said.

Not long ago, the White Sox thought they had the foundation of a championship team already in place. They had a formidable duo at the top of the rotation in Sale and Quintana, a proven closer in Robertson and powerful sluggers in Frazier and first baseman Jose Abreu.

The White Sox felt vindicated when they opened 2016 with a record of 24-14, but they quickly fell apart, going 54-70 the rest way, prompting their current trajectory.

By the time Hahn dealt Sale and Eaton, Frazier said he knew that if the White Sox didn’t jump out to a hot start, “Some things are going to happen.” Once Quintana was traded, Frazier recognized it was only a matter of time.

“I said, ‘All right, who’s next?’” Frazier said. “It was a flip of a coin between a couple guys.”

It turned out that the problem wasn’t with Chicago’s top players. The problem was with everybody else, something Hahn described as a “lack of organizational depth.”

With few reinforcements in the minors, the White Sox tried—and failed—to add talent around their core. It resulted in one of the most top-heavy teams in the majors, giving them a true stars-and-scrubs roster.

In the past, that might have worked. Not anymore.

“Look at the Cubs’ roster and the Astros’ roster, teams that have had recent success: They had depth,” Hahn said. “There’s a lot of premium talent on those rosters and they have important depth throughout the system where there isn’t much of a dropoff.”

Chicago White Sox general manager Rick Hahn chats with a player during a workout. Photo: Ross D. Franklin/Associated Press

In Hahn’s ideal world, the White Sox might have gotten here sooner. He acknowledged that there have been other moments where he and Williams proposed a total rebuild, as the team heads toward its fifth straight losing season.

A few years ago, the White Sox fan base might not have been quite so understanding. The same goes for longtime owner Jerry Reinsdorf, who at age 81 is quick to remind Hahn that he “is in a little different position in terms of the patience he has to show in the short term,” Hahn said.

The difference now is that the idea doesn’t sound so crazy. Fans are more receptive to this approach than ever before, Hahn said, because they “have seen the success of similar such processes around the league.”

So this time, the White Sox tore everything down and started over, a clear sign of a change in attitude. The Astros and Cubs opened the door for the rest of the league—and the White Sox waltzed right through it.

“We weren’t sure how it was going to be received,” Hahn said. “Even if, for whatever reason, fans didn’t take to it, we believe it’s going to be short-term suffering for long-term benefit.”

Write to Jared Diamond at jared.diamond@wsj.com