Actually, the NFL did have an "exhibition bowl" at the end of every season in the 1960's. The runners-up in each division (back in the two-division NFL days) played in what was officially known as the Bert Bell Memorial Playoff Bowl. In 1967, when the NFL moved to a four-division setup, the losers of the league semi-finals played in the game. The game was set for the week after the Championship game, and the week before the Super Bowl.
That game was called, in the most "polite" terms, the Toilet Bowl. The Detroit Lions won three of the 10 contests.