That's not necessarily true.
If you break OUR Strange's hands, he will become the Sorcerer Supreme. But the Strange in "What If...?" was emotionally a different man. The major difference is seen in how he had embraced an loving romantic connection ours did not. The point of variance isn't in the death of his girlfriend, but earlier, before the show, when he opened his heart to her.
It is possible that, with that connection, broken hands wouldn't lead him on that journey - he'd have someone else to lean on, and not have quite so much drive to regain what he had lost, but instead find a way forward with his damaged hands.