The intuition is actually simpler in my opinion. Modeling the likelihood in a non-parametric fashion is basically density estimation. The problem is that density estimation in dimensions higher than 2 is a problem well known to be difficult. Especially since you need to accurately estimate the absolute probability density values over the whole space, which need to be globally consistent. In contrast, the score only cares about relative density values, which are local properties. So that's an easier problem. However, you now need your local information to cover enough space, which is done through annealing/tempering.