Cool topic, lots of interesting thoughts.
I think the answer, like many before me mention, is somewhere in between. If you simply pick character guys, you're missing out of talent that could potentially be a great value. However if you simply look to employ those will injury or behavioral issues, you run the risk of shortened careers and self-implosion.
i think the pats do actually do this as well. The difference being they can recognize when it isn't working (I.e, Collins, Jones, etc) and can get somebody return value for taking the risk. Therefore, thy siphoned the success out of these character guys and unload them right before they are due huge pay days or are too damaging to the team.