Every great quarterback has a defining characteristic.

Tom Brady, even at 43, still excels in big games. Aaron Rodgers and Patrick Mahomes, with their prodigious arms, complete throws others wouldn’t dare attempt. Peyton Manning, a presnap savant, could decode the most complex of defenses.

Many will never come close to knowing what such excellence feels like, in any field. But when it comes to Drew Brees, another member of that exalted group of quarterbacks, trying to understand what distinguished him as he retired Sunday, exactly 15 years after he signed with the Saints — that feels a bit more accessible: Just grab a toothbrush and some toothpaste.

“I’ve challenged people to do this before,” said Zach Strief, a former offensive tackle who helped protect Brees for 12 seasons in New Orleans. “Brush your teeth with 275 strokes tomorrow. Do it that many times, then try to repeat it for 20 years. That’s how he lives his life. His attention to detail is his superpower.”

Over those 20 years, as Brees overcame a career-threatening shoulder injury to become one of the most statistically productive quarterbacks in N.FL. history, he trained his body and brain for optimal performance.

Because he couldn’t dislodge his head from his shoulders to see over towering linemen, the 6-foot Brees often threw passes blind. “You just see a ball appear out of nowhere,” said the former receiver Lance Moore, who played eight seasons with Brees.

Brees knew the coverage, the routes and where the ball was supposed to go, so it didn’t seem peculiar. It’s why every repetition in practice had to be perfect, and if it wasn’t, Brees and his receivers would stay after — communicating that need telepathically — until they aced it.

He reviewed the entire game plan after Saturday walk-throughs, drilling his cadence and progressions, dropping back without holding a ball, toiling alone in the Saints’ practice bubble. He arrived at the team’s training facility at 6 a.m. even if he played the night before. His wife, Brittany, would bring their children over at a certain time, and Brees would chase them around for a certain amount of time, and then they would leave at a certain time, so he could retreat to the darkness of the film room.

“It’s unnerving at first to watch him as a young player, because you’re like, ‘Damn, how do I replicate this?’” said Marques Colston, a Saints receiver from 2006 to 2015. “It put you in a mode where you had to match his intensity.”

Brees and Colston joined the Saints within weeks of each other in 2006. New Orleans drafted Colston that year, but Brees, after five seasons with the San Diego Chargers, chose the city. Identifying with its resilient spirit, he signed with the Saints to rebuild — his shoulder, his career, the organization, a region reeling from Hurricane Katrina.

With those projects long complete, Brees, 42, leaves the game after 20 years of unbridling his superpower to maximum effect.

“Over and above his outstanding performance, Drew came to represent the resolve, passion, and drive that resonates not only with Saints fans and football fans but our entire community,” Gayle Benson, the team’s principal owner, said in a statement.

When Brees arrived, the Saints were a woebegone franchise coming off a 3-13 season in 2005, with one playoff victory in 39 years. Brees reached the N.F.C. championship game in his first year, delivered a Super Bowl in his fourth — beating the Hall of Fame quarterbacks Kurt Warner, Brett Favre and Manning along the way — and won seven division titles, including in each of the last four seasons. He transformed the national perception of the Saints and recalibrated locals’ expectations of offensive proficiency.

When Brees arrived, New Orleans was recovering from the devastation wrought by Katrina, so much so that after Coach Sean Payton got lost while showing Brees around the area on his free-agent visit, driving past ravaged communities, he figured Brees would sign with Miami. Instead, Brees settled in Uptown New Orleans, restored a century-old home, and committed to raising millions of dollars to refurbish parks, schools and athletic fields.

When Brees arrived, his surgically repaired right shoulder was still ailing, and all throughout training camp and into the preseason his passes wobbled. Some teammates wondered whether he would ever recover. Payton did, too.

As Strief remembers it, Brees went to throw a 20-yard out route early in the Saints’ third preseason game, and his pass skipped 5 yards short of the receiver. Payton asked the quarterbacks coach, Pete Carmichael, who coached Brees in San Diego, “Is this as good as he gets?”

“I remember standing there thinking, like, oh wow,” Strief, who was hired last month as the Saints’ assistant offensive line coach, said. “Like, asking myself: ‘He’s an N.F.L. quarterback. How is that possible?’”

As Strief discovered, Brees progressed at his own pace. Meshing with Payton, he threw for 4,418 yards that season, the first of seven times he led the N.F.L. in that category. No one has completed more passes or thrown for more yards, and only Brady has thrown more touchdowns.

Some of Brees’s totals are bloated by the era, facilitated by rules changes, schematic innovations and a short-passing ethos. But in many years, the Saints needed Brees to throw just to offset their horrific defenses: Each of the five times New Orleans finished in the bottom seven in scoring defense, Brees led the league in passing. Over the last four seasons, as the Saints leaned more on their running game and a strong defense, Brees reinvented himself, throwing (even) shorter passes and fewer interceptions, never reaching double digits in that statistic after throwing 15 in 2016.

“You just knew the ball was going to be perfect coming from Drew Brees,” the former All-Pro cornerback Aqib Talib said in a telephone interview. “He’ll just find ways to kill you.”

Consistent as Brees was, sometimes that focus blinded him from change swirling around him. Long a vocal supporter of the military, he equated kneeling during the national anthem to protest police brutality with denigrating the flag.

As civil unrest roiled the country last summer, and as the league and its players grew more proactive about addressing systemic racism and social injustice, Brees reiterated that he considered it disrespectful to kneel. His comments angered teammates past and present, many of whom were mystified that someone generally so aware could be so insensitive. Brees later apologized, saying his comments “missed the mark.”

“It hurt — like, dang, Drew, really? No way,” Moore said. “But sometimes it takes a situation like that for somebody to grow. I’m not going to allow something like that to erase the history we had together. I had to help teach him a lesson, and I think it was a moment of reflection for him.”

Brees had ample time to ponder his future after the last three seasons, which all ended with a playoff defeat at the Superdome. Eliminated by the Rams in the playoffs after the 2018 season after officials missed a pass-interference call against Los Angeles, and by Minnesota in overtime after the 2019 season, when he missed five games with a thumb injury, the Saints lost to Tampa Bay at home in the divisional round in January in part because the Buccaneers converted two of Brees’s three interceptions into touchdowns.

That day Brees, already managing the aftermath of the 11 fractured ribs and punctured lung he sustained in Week 10, was also playing with — as revealed in an Instagram post Brittany Brees would make two days later — a torn fascia in his foot and a torn rotator cuff. Struggling to move the offense downfield against Tampa Bay, Brees passed for 134 yards, his fewest in 18 postseason games by far, and if it all seemed like a discordant conclusion to a career steeped in splendor, that’s because it was — but yet it still sort of misses the point.

So much of the Brees mythology focuses on what he lacks, things out of his control — the prototypical height of a quarterback, an Elway-esque arm, a second championship to enhance his legacy — instead of what he is, what he has, what he could do. And over the last two decades, as the N.F.L. transitioned into a passing league, no one summoned his superpower better to fulfill the position’s elemental responsibility — throwing a football accurately and consistently — finer than he did.