Overtime is not necessary to play when a game is not even after 4 quarters, so why should it be played when a game is tied after 4 quarters? Not being able to play until a winner emerges or forcing a winner to emerge is bad for the sport.
A tie is a legitimate result but because people, especially Americans are used to winners and losers, they play a maximum of 1 overtime period in the NFL of up to 10 minutes. It used to be 15 but cut to 10 in 2017. If score is tied after that, it ends as a tie.
You are playing overtime by arbitrary rules. You need to play the entire overtime, not modified sudden death. Even though there's been only 22 ties in the 44 year history of regular season overtime since it started in the 1974 season, there has been over 500 overtimes played. If those games were regulation scores, that's more ties and more legitimate teams making playoffs and not stumbling to a winner like you usually get in overtime. In chess, the goal is to win but sometimes my opponent and I play to a draw by not having enough pieces to force a win. We shouldn't use arbitrary rules just to have a winner when we proved on the board we were even. Ties happen. It's part of the game.
Arguments for overtime
1. If you don't allow a playoff game to end in a tie, then why allow it in the regular season?
Well there's an easy answer. In the regular season, your goal is to be good enough to have an entrance into the playoffs. In the playoffs, your goal is to be the best of a particular match-up. Regular season, you try to have an entrance. Playoffs, you play for advancement.
2. Every game should produce a winner and a loser.
Really? Then why over the 98 year history of the NFL that every game was not that way. Yes, a game usually has one and that's the goal of each team is to have a winner but why should a team play 80 or 90 minutes just to force a winner. We need real winners, not fake winners. Sometimes you have to shake your opponent's hand and recognize that neither team was better. No shame in a tie.
Arguments against overtime
1. A tie is a legitimate result
Yes it is. Just like a win and a loss is a legitimate result. Just because 2 teams share a result, doesn't make it incomplete. Football is a 60 minute sport and should stay that way. It can also help you make the playoffs. If the Saints, Eagles and Redskins are all 9-6 and the Eagles and Redskins hold the tie-breaker over the Saints, then the only way for the Saint to make the playoffs is win and hope the Eagles-Redskins game ends in a tie because if there's a winner between them, they advance over the Saints. So a 10-6 Saints team would beat out an Eagles and Redskins team that is 9-6-1.
2. It generally forces a team to get a less than fair result
Why should a team who is 99% as good as you get an excruciating and devastating loss just because you can't admit that you were equal to your opponent? A tie is the most fair of all results. By playing 60 minutes, you proved to them that both were worthy of the tie. By forcing a team to basically win, you are giving them a more unfair result and also your opponent too.
In case you are wondering but other sports like baseball and basketball plays to a winner, you're right to a point. They play real baseball and real basketball to decide it and they don't beat each other up like in football. However, Japanese baseball leagues play a maximum of 12 innings, so ties are possible, although unlikely to have more than 5 or 6 in a 144 game season. Playoff games must produce a winner because it's a tournament more or less of a 1 and done season each game but regular season games are based on the body of your record.