For once, a player had Bill Belichick over a barrel. And the best coach in football, the one who notoriously eschews paying premium salaries to players, gave in. So in 2015, Belichick made Stephen Gostkowski—the New England Patriots’ kicker—the NFL’s highest-paid player at his position.

Belichick and Tom Brady represent the Patriots’ excellence, but Gostkowski who explains a less appreciated aspect of the franchise’s stability. While other teams suffer through a succession of kickers—often quickly replaced after an iconic...