Rod the Bod is an interesting but probably unfeasible candidate. Let's say there's an actual chance he leaves the team he's an icon of. Rod has had much more playoff success than Keefe, has won a couple of division titles over him, and has finished higher in the overall standings pretty consistently. The Hurricanes also always dominate the public advanced stats and they are one of if not the most consistent team in terms of playstyle.
There are a few things that are "Leafy" with him and his Canes. Scoring issues have cropped up in the series' they've lost, same as us. And despite their success they've won the same amount of Conference Finals games as us under him, zero. In the postseason they’ve been plagued by a lot of one goal losses, hot opposition goalies, poor special teams, and bad bounces. Again similar to us only they plague Carolina in rounds beyond round 1.
Brind'Amour has gotten outcoached by coaches who've taken Keefe's (and Babcock's) lunch money over the years. Cassidy, Cooper, and Maurice, though he did get Cassidy once two seasons ago, but so did Keefe catch Cooper once. As far as the other coaches he's faced and beaten, Reirden, Trotz, Quinn, Hynes, Lambert, Ruff, and Roy, only Trotz really sticks out as impressive. In a division like the Atlantic with the likes of Cooper, Maurice, and Montgomery, would he have had as much success over the years if he was consistently going against these guys instead of some of the mediocrity in the Metro? Again, not to take away from him and Carolina but it's worth noting that a lot of their series wins have been against some pretty average coaches (outside of Trotz and Cassidy). Laviolette is also currently dismantling the Hurricanes.
This doesn't even get into the financial side, with rumours of his next contract being bigger than Babcock's I'm not sure how much of a stomach MLSE has for yet another big time hire in the leagues most volatile position off the ice. Yeah the Leafs make money hand over fist but the Raptors aren't competitive right now and TFC and the Argos lose more money than they make. Plus one of the recent big issues among the heads of MLSE is the Rogers side being pissed about having to pay Babcock for so long after he was fired. Who knows if they'll want to commit that much money to a coach again when you've got guys with better playoff success kicking around in Berube and Gallant who will probably cost less.
If the Rangers can finish the job on Saturday and sweep Carolina the Leafs should definitely check in. If Brind'Amour could bring the same level of consistency he has brought in Carolina maybe the Leafs higher level of elite players (specifically Nylander and Matthews) could help both sides with the problems that have plagued them come playoff time.