Intuitive way to understand Gauss-Bonnet Theorem

There are quite a lot of intuitive explanations for Levi-Civita connection, exterior derivatives and other concepts in Differential and Riemannian geometry. But I couldn't find any resource with an intuitive explanation of Gauss-Bonnet Theorem. The proof I am aware of (given in John Lee's Riemannian Manifolds) seems like a trick of notation and usage of Stoke's theorem. I tried to come up with a more visual way of thinking about this theorem using Gauss map but so far wasn't able to do so.

What is the intuition behind this theorem? How did Gauss come up with it himself? (If it was him) So far for me, it seems that the way one would come up with such a theorem is to notice the pattern for several shapes, write down the formula and try to prove it using the available techniques. Yet, It seems unsatisfying.

All references are welcome.

Thank you.

Solution 1:

This will surely not be satisfactory, but this is an attempt, a preamble to the proof of the Gauss-Bonnet theorem in the text I coauthored, Discrete and Computational Geometry. The key intuition is:

if we dent the surface, saddles must emerge so that the amount of negative and positive curvatures cancel out perfectly