None of the available coaches directly scream long term success honestly.
Karl prefers a 30-30-30 system. He favors around 30 layups, 30 free throws, and 30 assists each game. The system emphasizes constant ball movement and spacing without a select player holding the ball for too long or shooting 15-20 foot jump shots. However, that style often struggles in the slower-paced Playoffs, especially against teams playing zone or against teams with strong interior defense. The "others" struggle, and the players that have to take over in those moments are offensively-elite talent like Melo or Kobe.
In terms of defense, Karl's never been a strict tactician of it. He does emphasize it during practices, which usually consist of 60% defense and 40% offense. However, his teams have usually been average to bottom half in defense. So when your offense is negated in the Playoffs and your defense wasn't your strongest suit to begin with, his teams are going to struggle in the Playoffs. The question is...can he properly adjust? Out of the 22 times he has made the Playoffs, 14 of those have been first round exits.
Lastly, in times of team struggles, Karl has been known to exhibit negative body language. During the games, he'll just sink his head into his hands, slouch down, walk slower than usual without enthusiasm during timeouts, et cetera. During postgame interviews, he'll constantly show frustration and often mutter not the best remarks.
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Scott primarily relies on the Princeton Offense, which emphasizes constant motion, passing, cuts, teamwork, and a relatively high basketball IQ. It's a tough offense to guard, but at the same time, it's difficult to learn and constantly stay focused each play. Scott has always valued defense and rebounding, and the Nets team he coached to the Finals were top 5 in defense. His New Orleans teams ranked anywhere from 6th to upper half in terms of defense. It's important to note that he's been successful defensively when given a strong perimeter (Kidd/CP3) and interior (Martin/Chandler) defender.
Scott can often let his ego get in the way and refuse to change his system or philosophy. When his Nets team was getting hurt inside during the Finals by the talents of Shaq and Duncan, he constantly refused to throw double-teams. He's often placed key players on the bench for too long without effectively communicating with that player over why he was benched.
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Whoever is ultimately hired, I can see various complaints during the season and/or Playoffs. Whether it be Karl's Playoff struggles or Scott's ego and hence inability to properly adjust. I still wonder why we didn't hire Phil...Or at least Brian Shaw who could continue coaching the Triangle system and seek assistance with Phil. Well, I hope the Lakers FO is thinking long and hard this time.