I'll try to keep this as succinct as possible.
There is a good argument that this Leafs team was the worst defensive team of the modern era. They allowed 36 shots per game. That's 2945 shots this year, which is the most EVER in a 82 game season.
Here's the proof.
So this isn't just your normal bad year. This was a historic level of futility in terms of preventing shots against. There are two possibilities. Either the Leafs roster is the worst collection of defensive players ever, or they had terrible coaching. I find the latter far, far more likely.
Carlyle is simply a dinosaur trying to compete in the modern NHL. He has two huge philosophical problems. The first is that after faceoff losses, he has his players back up into a defensive shell. This despite the fact that all the smart teams forecheck hard after faceoff losses, because possession of the puck is the most important part of the game. Carlyle essentially concedes possession and the team just backs up and waits for the opponents to attack.
The second problem is Carlyle's love of the dump and chase, a popular strategy in the past that all the smart teams have stopped doing. Again, possession of the puck is the biggest factor for success. The Leafs are freely giving it to the other team and hoping to get it back. It's stupid. Teams like LA, Chicago and Boston don't dump the puck, they hold onto it as much as the possibly can.
This isn't about players not having enough compete level or any BS buzz words Leafs management always throws around. It's coaching tactics, plain and simple. The evidence of this is clear. Check out this graph to see how Anaheim has significantly improved since firing Carlyle, while the Leafs have gotten worse since hiring him:

People always bring up Carlyle's Stanley Cup to defend him. One title doesn't mean he's a great coach. He won in his first year, before he had much of a chance to implement his system. He inherited a team with Scott Neidermeyer and Chris Pronger, two of the best defensemen EVER. It would be impossible to be terrible defensively with those two. The team continually got worse under Carlyle until they fired him. Since firing Carlyle, Anaheim has magically gone from mediocre to being a contender in the West.
The main difference between the Carlyle and Ron Wilson (or Paul Maurice, for that matter) teams is goaltending. Bernier posted the 6th-best save percentage at even strength in the league. He was great, especially considering the absurd and historic amounts of shots he faced every night. That performance masked the underlying and fatal flaws with the team. It's very likely that Bernier is not as good next season. If that happens and the Leafs continue this path of allowing so many shots, they will probably have one of the worst records in the league.
Not only is Carlyle making the current players worse with his tactics, but he's making the roster worse with the help of Nonis. Carlyle doesn't like Grabovski, so he's bought out. Meanwhile Grabovski was Washington's best possession player this year. Carlyle doesn't like MacArthur, so he's let go. Meanwhile MacArthur was Ottawa's best possession player this year. Now the players Carlyle doesn't seem to like are Kadri and Gardiner. Surprise, surprise, they're the Leafs two best possession players right now.
The data for that is right here.
Ignoring possession stats is essentially like ignoring OBP in baseball and seeking out hitters who make the most outs. It's proven to have the best correlation to winning, but the Leafs seem to actively go against all the data. It's madness. Trading our two best possession guys in Kadri and Gardiner would be ridiculous. Of course it's what I expect, since the Leafs have made stupid decisions pretty consistently. But we need more guys like them, more skilled guys instead of the grinders Carlyle prefers.
I find the idea of retaining Carlyle because they're waiting for Babcock, or McLellan, or whoever, to be completely ridiculous. You don't keep a coach because you're waiting for the perfect match. You fire him because he's incompetent. My first choice for a replacement would be a guy like Davis Payne. He's the lead assistant in LA, the guy responsible for their strategy and integrating their analytics into that. The Leafs don't need to act like the stupid Knicks. We don't need to make splashy, headline-grabbing hires. Steal people away from smarter franchises. If not LA, target someone from Chicago or Boston. They're doing things right. We're doing things wrong, it's time to change.
The Leafs have been one of the worst teams in the sport since the 2005 lockout. It's because that lockout created a system that rewards smart franchises. Suddenly we couldn't spend our way into contention like baseball teams, it was up to management to make prudent and intelligent decisions. They're consistently failed to do that for almost a decade. I was naive to ever think this time would be different.
My two favourite Leafs writers put it all more poetically. I suggest everyone read the columns by
James Mirtle and
Bruce Arthur.