What was the date of the beginning of the “ modern” era?
It's Green regardless if we start the clock after the Lombardi years, but there are a few ways to define "modern".
1) This one takes in a much longer historical sweep than most would consider, but the 1958 Championship Game, "The Greatest Game Ever Played", is widely regarded as ushering in the TV era. This had less to do with how the game was played or things counted as lauching the league into broad public consciousness as TVs were entering homes en mass. In 1950, 20% of households had a TV, By 1960, it was 90%.
2) Some folks like to put the dividing line at "Super Bowl Era" but more frequently you'll see historical stats quoted from 1970 forward, the first year of the NFL-AFL merger being consumated, with the AFL teams integregrated into the NFL schedule.
3) More idiosyncratic, but maybe more to the point, would be 1978. The schedule was expanded to 16 games. The rule changes (5 yard chuck rule and a more liberal interpretation of how O-Lineman could use their hands), opened up the passing game. This was one year after the "Deacon Jones" rule went into affect where head slapping an O-Lineman became illegal. 1978 set the path, with more rules layered on top to favor the passing game.
If one were to point to one watershed year where both statistical measures and the way the game is played changed dramatically, this would be it. Now, you didn't see teams suddenly start throwing the ball 600 times per season as is fairly common today. It was gradual as a new generation of schematic concepts rolled in along with a new generation of players with different traits and skill sets who could execute new techniques. First, "Air Coryell", followed by Walsh's "West Coast" in Cincinnati before it moved to San Francisico. Maybe 1978 marks "post-modern" football.
To illustrate a bit of this, I think that's Carl Eller below executing a head slap, maybe more of a punch, on an OT trying to block him with his forearms, while what looks like Rodger "The Dodger" Stauback assessing his chances for survival instead of looking downfield. There might be some video out there showing Mel Blount mauling receivers all the way through routes. You could do that until the ball was in the air directed at that player so long as you didn't hold.
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4) After fumbling around with Plan B free agency for a few years, 1993 ushered in true unrestricted free agency with the salary cap following in 1994. While those changes did not change the way the game is played directly, it surely upended the sleepy player "ownership" for life of the reserve clause and changed radically how rosters are constructed. But as for the question at hand, career team statistics have become increasingly less relevant as players hop to the highest bidder after 4 years. "Post-post modern"?
In the end, what is "modern" changes with every generation as what was once a watershed is forgotten to time. I would guess most people reading this were not alive in 1958 and many others don't realize the significance of those 1978 changes.