Win the most Championships/Super Bowls? Most fourth quarter come-from-behind victories? Most game-winning drives? Greatest statistics? Division first place year after year? Elevate the rest of the team players? Field team leadership?
When a sportswriter says "Quarterback X is the Greatest of All-Time", it would be interesting to know (A) their definition of "Greatest", and, (B) how that was measured, and, (C) are quarterbacks before the Super Bowl era considered.
What if a great quarterback is surrounded by a lousy team year after year (such as Dan Marino and Archie Manning)? What if a marginally decent quarterback (such as Terry Bradshaw) is on a team with an abundance of outstanding future hall of fame players? What if an average (or less) quarterback backs up a bold prediction about winning his only Super Bowl (such as Joe Namath)? What about someone who probably would be an top QB if he was not a backup QB (such as Jeff Rutledge). What if a seemingly outstanding quarterback has bad coaches (such as Todd Blackledge)?
Can a great coach make a good quarterback look great? Do great quarterbacks make good receivers (Reggie Wayne) and good coaches (Bill Belichick) into great receivers and great coaches? Or, do great coaches (Paul Brown, Vince Lombardi, Joe Walsh) and/or great receivers (Jerry Rice, , Randy Moss, Lynn Swann) make good quarterbacks look great? Then there is that entire argument about quarterbacks from different eras with different rules that favor quarterbacks more, or less, than other eras.
Quarterbacks who play well for 16 or more seasons tend to rank in the top mainly because they had a chance to throw for more yards and TD's and win games and championships. True, they did have to have many good seasons to get to that point. But, it does move them above most everyone else for the stats. Maybe a player who accomplishes great stats and many wins in ten years or less, including winning the championship, should be considered with more weight.
Interestingly, when looking at stats "per game" or "per pass attempt" or "per season", the players from the early years (40's, 50's and 60's) usually dominate the top ten per statistic. Same can be said for such things as making fumbles. What does that tell us? Unfortunately, there are some records that were not kept on QB's until the latter years (e.g. sacks).
Based on six different ratings here, the short answer of The Greatest Professional Quarterback is Otto Graham (played 1945-1955). Only a few quarterbacks even come close to what he did. Tom Brady (who has already been coronated as such by many modern sportswriters who seemingly know too little about Graham and/or ignore pre-Super Bowl times or only count total career numbers) and Patrick Mahomes are the closest. The below will show why Graham is THE G.O.A.T.!!!
Given all of the above, the below accompanying very large spreadsheet is an attempt at looking at these various factors. Each quarterback has the various objective statistics as well as some subjective indicators. Additionally, to make sure that the very best will rise to the top of the list, over eighty quarterbacks are considered. Finally, five different ratings are calculated per quarterback. Each rating is weighted towards one factor or another.
Interestingly, the more modern quarterbacks seem to be favored more by these rating systems. That could be because of rule changes favoring quarterbacks (which allows better and better statistics) and because athletes simply know how to better train and prepare than in years past, plus, they certainly have better medical care and higher salaries (Graham's highest salary is what in 2022 would be about $250,000).
To help with that bias, one of the ratings is based purely on averages and percentages -- when a QB gets the ball, he either throws it, fumbles it, runs with it, gets sacked, or hands it off. These stats ignore the handoff. The other three possibilities can be a completed (and maybe a TD) or incomplete pass, a lost or recovered fumble, yards gained (and maybe a TD) or lost by running or was sacked. The combination of what the QB does over time adds up to how successful the team is at winning.
When all ratings are taken together, the same few quarterbacks consistently rank in the top five (although the one only using averages and percentages really shakes things up!). And, surprisingly, the same single quarterback always rates in the Top Two of all stats rankings.