Without the constraints of a salary cup then yes Doughty may well be what leads TOR to the cup. However his next deal will be 11 or 12 mill, and Matthews will probably get 12 as well. You can't win in a cap world with 24 mill tied up in 2 players

I don't know why some Leaf fans and other hockey fans, think this missing # 1 Dman is suddenly going to lead the Leafs to the Cup. There really is no logic behind it. Just checking the last ten games the Leafs are scoring just 1.5 goals a game, so the Leafs have holes to their team that one player can't solve. This # 1 Dman is the solution? Doughty missed the playoffs last year, Hedman missed the playoffs, OEL always misses the playoffs, Karlsson three playoff round victories in eight years. So it's kinda fantasy that this # 1 Dman, is the panacea for all problems. And if the Leafs were to trade for Doughty (or others), you can be sure it would weaken other parts of the team.
I don't know Doughty. But LA is his home. He really doesn't want to leave.(I would think) If a team were to pay so much more that Kings, yea maybe he signs with another team in the summer of 2019. But do you really want to be the team that overpays for UFAs? We have seen how well that has worked out throughout the years.
To answer your opening statement " Could you see the Leafs winning the Cup with Doughty on the team?" Yes. Also, No.
Here's a question. " Could you see the Leafs winning the Cup without Doughty on the team?" Yes. Also No.

I can see that logic. Leafs fans have never been closer to winning a cup in decades. We've had decent to bismol teams. We don't know how to buy at the Deadline, we're usually on the other side of the conversation. If it was up to me, I'd sell JVR and Bozak, maybe Leo too, resign Will now Matts and Marner ASAP, Try to trade Marleau next year at free agency or the year he has 1 year left, to bide time for our youngens like Lilegren, Dermott, Kappy and Aaltonen to develop.