They also picked up Ray Emery, who was essentially a pariah in the NHL at the time for his locker-room behavior.
It's a case by case basis, and I doubt a player's previous behavior has as much impact on their decision as a) can they help the team, b) do they understand this is a fresh start, and c) whether they feel the leadership group can straighten them out.
A lot of what you say is right (case by case scenario) ... but there's a huge difference between Emery and Drouin. Emery's problems stemmed from a terrible temper on the ice. He was a fighter who took it too far. By the time he got to the Hawks, Emery worked his butt off from a career threatening disease ... even being nominated for a Masterton multiple times.
Drouin on the other hand is purposely sitting out ... demanding a trade because he feels entitled to a top 6 position on a Stanley Cup caliber team. It's been a lot of off ice drama with Drouin.
The Hawks do have the lockerroom presence to mentor Drouin. But it won't be much different from the Lightning. He's going to have to EARN time on the top 6 ... something he wasn't able to handle at Tampa. Emery on the other hand had a tryout with the Hawks and EARNED a contract. And Q never has a problem with players who'll work for ice time.
That being said, I wouldn't mind Drouin on the Hawks. But the risk / reward has to be appealing. Yzerman has to give us a discount for the Hawks to even consider the move. All indications are that the Lightning want a bounty.