All teams are trying to injure players, imo. The Saints were just more outgoing about it. The NFL is a violent sport and delivering big hits is the goal of any defensive player.
This has been blown out of proportion. Hannibal Navies wasn't a dirty player but he put Smith on IR for a year. While the Saints were being paid to do as much and didn't do damage anywhere near as close.
First, I do not think that all players are trying to cause injury to other players and you are flat out wrong that all teams are trying specifically to injure players. I do think they want to hit them hard and intimidate them. However, even if all players on some level have an intent to injure (something I actually do not agree with), it is certainly not true that all organizations have an institutionalized bounty program like the Saints did. We should be clear here that these bounties were for more than hitting someone so they got up slow, it was for knocking people out of games... making them unable to finish games etc. These were more serious than slowing someone down with a hard hit.
This is not blown out of proportion. The Saints coaches propped up a system that targeted specific players and awarded specific amounts. It flew in the face of all of the recent rules the NFL has been struggling to implement RE: player safety. It also violates rules relating to contracts etc. Finally, the organization itself basically lied to the NFL about it. All of these things are bad, but in varying degrees.
Obviously the first was one of the worst. It should be clear to anyone why it is important that the NFL present an image that it is safety-conscious, even if the players themselves prefer to act like THEY are not. Individual players are unlikely to face law suits from former players over safety concerns like the NFL is. On top of that, players themselves need to be concerned with this stuff because they already have generally relatively short careers, and the risk of injury is already high without some defensive coordinator putting your picture on a board and telling people they can make $10k for breaking you in two. The intent here is what was the problem.