High dimensionality allows gradient descent to make progress independently in each dimension. When you get stuck in one dimension, you make progress in another

High dimensionality allows gradient descent to make progress independently in each dimension. When you get stuck in one dimension, you make progress in another dimension until you can progress in the previously blocked dimension. Also, even if you do get stuck in X dimensions, you are unlikely to significantly reduce that X by finding a better starting point and if there is redundancy, then it doesn't matter that you have the local minima in one dimension but not the other dimension. The end result is that the quality differences between the local minima end up as insignificant. They are sufficiently good compared to the global minima.