It's obviously not completely fair to compare Bradshaw to more modern quarterbacks since the rules today heavily favor the passing game, but Bradshaw's career stats are pretty ugly by any objective measure. Roger Staubach played in that era and has much better stats. Bradshaw threw just two more career touchdowns than interceptions. I don't care what era you play in, that's bad. However, he did up his play BIG TIME in the postseason and particularly the Super Bowl. He also certainly had a big luxury in that he that defense to back him up.
Montana has the stats AND the Super Bowls. Brady too. Your Eli's and Bradshaws are certainly HOF quarterbacks because they played so good when it mattered most. But they don't deserve to be ranked above guys like Manning and Marino, who played superior football at the position.
Where rings come into the picture is when you're trying to compare guys like Manning and Brady who are so similar in most other areas (reg season stats, reg season wins, division titles, etc). It's never going to be completely fair because it's not like a quarterback can play defense, but it's certainly an important thing to look at. At the end of the day, it's just something that is debated by talking heads and people on a message board. It's not like Manning and Brady are fighting for some "GOAT prize" at the end of their career.
That being said, you can't downplay the importance of this looming AFC Championship Game when it comes to legacies.