No team has ever beaten all five bird teams in one season. The Bird Gauntlet is extremely hard to accomplish, and only possible for the twelve non-bird NFC teams. It requires a team to have an excellent year and the perfect schedule, or a lot of luck with playoff seedings.
Due to the yearly divisional matchup cycles, every twelve years Washington, New York, and Dallas are guaranteed to face the Eagles, Ravens, Cardinals, and Seahawks, and one of them will face the Falcons if they finished in the same place within their respective divisions the previous year. This happened this past season, with the Cowboys facing all five bird teams, but they could only defeat the Falcons and Eagles.
The same is true for the Panthers, Saints, and Buccaneers in the NFCS, except swap the the Eagles for the Falcons. Whoever finishes in the same spot as the Eagles in 2021 will get the opportunity in 2022.
The Rams and 49ers play the Ravens, Seahawks, and Cardinals very four years, but have to finish in the same place as the Eagles and Falcons. or face them in the playoffs.
The NFCN teams have the slimmest chance of the ideal schedule, as they play the Ravens, Cardinals, and Seahawks every twelve years, but have to finish in the same place as the Eagles and Falcons, or face them in the playoffs. Despite the odds, the Lions did exactly that, and will go against all five teams in the regular season. Any other Bird Gauntlet hopeful will have to get very lucky with playoff seeding, and outside the NFCN they'll have to knock off the Ravens in the Super Bowl.
Three teams have come very close, beating all the bird teams save one: the '03 Rams (Eagles), the '08 Giants (Falcons) and the '15 Panthers (Ravens).