Randy Carlyle was fired before this season even began. He was fired before he even got his extension. If you don't believe that, you don't understand how NHL front office changes work. When a new team President and/or GM are announced, you can bet your ass that unless that team has an iconic coach, which Randy Carlyle is not, that man will bring in his own people at some point. In most cases, they simply hold onto the "bullets" of firing these men in the event the team they take over is still stumbling.
If a new President is named, for example, and immediately cleans house, and the team struggles, that's on him. If he's hired and fires no one and the team struggles, he has a few bullets in the clip to let loose so that the blame can still be shouldered by the men he fires. The men he then hires, and the fate of the club from that point on are on him.
Carlyle is a terrible coach. I doubt he coaches in the NHL again. The real key now is who the next guy is, because it won't be Peter Horachek, who is going to be named interim coach. Horachek is a lame duck to ride out the rest of the season (because he was already an assistant and is under contract).
They'll wait until the summer, IMO, to see what happens with Babcock. If he hits the market, he's their man. They'll back the dump trucks up and just unload cash on his front lawn until he agrees. Otherwise, it's a wait-and-see scenario with Horachek playing lame duck until they pull the trigger on someone long-term.
If you ask me, they need to invest in something long-term here. Something that jives with the plan Shanahan has for where this Leafs team will be years from now. Torts, Bylsma, etc. don't fit that mold. They're "win now" guys. Maybe less so Tortorella, but that depends on what your expectations are in bringing him in. He did a great job of developing a work ethic culture among the Rangers' young players, but that flamed out big time with the construction of the Canucks' roster, which is similar to the structure of the Leafs' roster barring trades.
