As word of each NFL head-coaching hire landed this week, a sprawling network of black coaches and their advocates traded messages bemoaning what they saw as the latest in a long string of insults to their fraternity. One coach summed it up in a text to his agent: “I’m sad for all of us.”

Four of the league’s five coaching vacancies have been filled: Joe Judge in New York to the Giants, Mike McCarthy in Dallas, Matt Rhule in Carolina and Ron Rivera in Washington. The reported front-runners for the fifth opening, in Cleveland, are both white. If the Browns job goes to one of them, it would leave the NFL, in which roughly 70 percent of the players are black, with just three black head coaches. That’s as many as there were in 2003, when the NFL instituted the Rooney Rule, which requires a team to interview at least one minority candidate when it has a vacancy.

This season has been especially frustrating, several coaches said. That’s because there was one candidate who was not just the best black coach available but, they said, the best coach on the market: Eric Bieniemy, offensive coordinator for the Kansas City Chiefs. Bieniemy has interviewed with seven teams over the past two years, including the Browns, Panthers and Giants this hiring season. If he’s hired by the Browns, he would be the latest in a string of former offensive coordinators under Andy Reid to land top jobs, after the Philadelphia Eagles’ Doug Pederson and the Chicago Bears’ Matt Nagy. If not, it will be viewed by many black coaches as the most prominent snub in recent years.

“People have gotten jobs because of Brett Favre, Tom Brady, Peyton Manning and, recently, Patrick Mahomes. But Eric Bieniemy doesn’t, and he followed the same path,” Redskins assistant coach Ray Horton told The Washington Post. “There’s the frustrating part.”

“Pederson didn’t call plays; Nagy didn’t call plays. But they’re both good enough,” added a prominent black offensive coach, who spoke on the condition of anonymity for fear of jeopardizing his own career trajectory. “Andy Reid, one of the winningest coaches in the NFL, gives E.B. his full endorsement, and you’re telling me that’s not good enough?”

“Watching E.B. get passed over has a big ripple effect,” a black position coach said, “because now you have guys who are questioning if there is even a chance to elevate in the NFL. You want the best coaching candidates, regardless of race. And if you’re biased against black coaches, you’re overlooking a lot of talent.”

As jobs got filled this week, viral clips of ESPN’s Stephen A. Smith and Ryan Clark excoriating NFL owners found an audience in group texts and email chains across the fraternity of black coaches. One prominent agent told The Post that a handful of coaching clients have even inquired about opportunities in college football, where the minority coaching numbers are similarly bleak but more jobs are available.

Two of the three assistant coaches who spoke to The Post — Horton and the offensive assistant — said they have been through a process familiar to Bieniemy and others: sitting through interviews and wondering whether the people sitting across from them were actually interested in their services or merely checking a box.

“Mike Tomlin wasn’t supposed to get the Pittsburgh job, but he got it in the interview,” the offensive coach said. “So there’s hope you can go in and win the room. At the same time, there comes a point when they start to think you’re a dumb n----- just for going. I’ve felt that, being there as a token, where you know you have no shot.”

“These owners want to win,” Horton said. And he said he understands that a team such as the Giants may believe “Joe Judge is going to take them to the Super Bowl. . . . I can’t argue with that.

“But for us, the red line in the sand keeps moving,” Horton said. “First they said, ‘Well, you have to be a coordinator.’ Then you had to be a coordinator who actually calls the plays. Well, now that line moved, and they want an offensive coach. And now all of a sudden it’s a wide receivers coach.”

The Giants hired Judge after he spent one season as New England’s wide receivers coach and five seasons coaching special teams before that. At 38, Judge has never coordinated an offense or defense in college or the pros. Rhule, 44, coached Baylor for three seasons and Temple before that.

That Rhule and Judge have top jobs — and Bieniemy doesn’t — riles advocates for increased diversity in the coaching ranks. Brad Childress, a senior offensive assistant for the Bears, was the Vikings’ head coach in 2006 when he hired Bieniemy to coach running backs. He has acted as a reference for Bieniemy several times, and he has a favorite anecdote to share.

“I watched him coaching Adrian Peterson like he was a walk-on,” Childress, who is white, told The Post. In Minnesota, Bieniemy barked instructions and criticism at the all-pro running back. Years later, during a practice for the 2016 Pro Bowl, Childress and Peterson watched as Bieniemy tutored another player. “That man is the truth,” Peterson said to Childress.

“I’m at a loss to explain it, and I don’t know what kind of change you can make to fix it,” Childress said. “If black coaches are frustrated, they should be. We all should be. You have to be conscious of diversity. You want teachers the mentees can listen to and relate to. If you don’t want that, you’re sticking your head in the sand.”

Childress recalled being in a room full of coaches when the Rooney Rule was introduced and listening to Herm Edwards, then head coach of the New York Jets, pump up the measure.

“But the thing hasn’t taken hold,” Childress said. “You can say all you want, but the numbers tell the truth. When you have a group of billionaires who have a make and model of what they think a coach looks like, there’s no telling them what to do.”

Childress said it’s possible Bieniemy and others in the minority coaching ranks — including San Francisco 49ers defensive coordinator Robert Saleh, who is Arab American — may have found jobs this hiring season if their teams weren’t still in the playoffs. NFL rules prohibit coaches from signing contracts with new teams while their teams are still competing.

But coaches who have sat through fruitless job interviews are skeptical of that explanation. For them, the lack of minority hires can be explained by racial bias, whether conscious or unconscious, among the league’s owners, none of whom is black. No trade organization, union, advocacy group or interview rule can fix that, they said.

“I’ve been in interviews and asked later, ‘Why did you hire him instead of me?’ And they’ve said they felt more comfortable with the guy,” Horton said. “How do you quantify that? Why did they feel more comfortable with him?”

“I’m so sick and tired of talking about this,” the black offensive coach said. “This is the world we live in. We just have to continue to pay our dues. I’m okay. Just keep paying me, and I’ll head out to pasture. There are a lot of coaches of color that have helped white coaches win Super Bowls, and you never see their faces. They never got out front, never got to show what they were capable of. This is the status quo.

“It’s not just Eric. Why isn’t Leslie Frazier good enough? Why isn’t Kris Richard good enough? Why isn’t Jim Caldwell good enough? America doesn’t want to see a black man as the face of a franchise.”