"We already know what to do from college football. Go second so you can know what you need. Problem solved."
This is the argument I saw most often about this decision, and it's just wrong. While there's certainly a benefit to getting the ball second -- let's call this being Team 2 from this point forward -- and knowing what you need, there are two major differences between the NFL's playoff overtime process and college football's. One is field position: The NFL's rule starts each possession with a standard kickoff, while college football overtime begins on the 25-yard line for two possessions before starting on the 3-yard line for the third overtime.
The other difference is more significant, and it's the factor Shanahan brought up in making his decision. In college football, teams always get a chance to either match or top the first team's (Team 1) score on their drive, so there's not much thought: Most teams will prefer to be Team 2 and know what they need if they want to win. With that being said, from 2013 to 2021, Team 2 won only 51% of the time in a 243-game sample of college football overtimes, so this isn't a significant advantage.
If the two NFL teams are tied after they each get one possession with the football, though, Team 1 gets the ball for the third possession of overtime with the opportunity to win by scoring a field goal or touchdown -- without giving the ball back to the opposing team. That's a dramatic advantage! Remember that from 1994 to 2011, when NFL teams were able to win in overtime by kicking a field goal and without needing to give the opposition another possession, teams that won the coin toss won the game nearly 60% of the time. If Team 1 gets to the third possession of overtime, it's essentially playing a game as if it had won the toss under the league's old playoff rules.
That 60% number also underestimates Team 1's chances of winning on drive No. 3 of overtime. The NFL was spurred to make the change in part by the improving performance of kickers on long-distance field goals, which has only continued. From 2000 to 2011, kickers hit just over 55% of their field goals in the 50- to 59-yard range. Over the past five seasons, that mark is up to 66.6%. The league has moved up the field position for touchbacks from the 20-yard line to the 25, adding 5 free yards for offenses. With defenses tired after a full game and another possession of overtime, it also should be easier to move the ball. Teams might need only 35 yards on that third drive to get into field goal range for a good kicker.
Make no mistake: Getting the ball third is an enormous advantage if a team can get there. The "if" is the hard part.