Carlyle is an afwul coach. His system (if he has one), simply consists of collapsing on defence and dumping and chasing. So, it should be no surprise that his teams have always been at the bottom of the NHL in terms of possession. I cannot be the only one who remembers all those games that we were stuck in our own zone and were outshot by an absolutely massive margin. We always had to rely on our goalies to bail us out.
He was an idiot and clueless as a head coach, as well. He would wonder why his teams were bad at puck possession and then he would always tell them to dump and chase, which is as anti-possession as it gets. The defencemen were always left to dry because Carlyle never would ask the forwards to help out.
He also ran good players like Grabovski and MacArthur out of town and heavily underused and misused them, while they were here. Two good, two way players, who were also good possession players. He didn't have any problem playing Orr and McLaren, though. He also didn't have any problem calling out our goalies, which is dumb, since our goalies were the biggest part of the little success we had.
Also, remember when he cost us a second round pick by playing Holland the necessary 25 games? Too bad Holland only played 5 minutes per game and on the 4th line.
The 'Carlyle got the Leafs to the playoffs" narrative is also way overstated. Yes, he got us into the playoffs, but only because it was a shortened season. That collapse in game 7 only happened because we were such an awful possession team. For essentially the entirety of third period, we were stuck in our own zone and had to rely on Reimer to bail us out. Unfortunately, Reimer became the scapegoat, even though he was the biggest reason we even made it to the post season in the first place.
The "Carlyle got the best out of his players" is another narrative that got old fast. Carlyle had us playing unsustainable hockey. That's it. Every year, we would collapse with him, but there were always people defending him for some reason. For instance, people would say that he had us in a playoff position when he was fired, but conveniently forget there were teams below us with a bunch of games. The Leafs also had an abnormal (and unsustainable) shooting percentage and Bernier was playing absolutely lights out. Not to mention the Leafs only won like 2 games out of their previous 9, when he was canned.
Therefore, because of all the reason listed, Carlyle is an awful coach, who has no place in today's NHL.
tl;dr: Carlyle is a horrible coach, with an equally terrible system. His system was extremely anti-possession, which obviously doesn't work in today's NHL. He made a lot of dumb choices as the head coach and ran good players out of town, while playing goons. He was lucky he got goal tending and a good shooting percentage from the team, because it masks a lot of his failures. His results were always unsustainable. He should not have a place in today's NHL.