Euclidean angles on curved surfaces

Question

So usually when defining angles between two curves on some curved surface, we essentially take the angle between their unit speed tangent vectors where they meet. This definition leads to nice things like the Gauss–Bonnet theorem and angles around a point sum to 360 degrees. If we now instead chose to define the angle between two curves by some version of the cosine rule, where the third side is also a geodesic, this leads to some pretty weird things of course. Obviously, the sum of angles in a triangle will always be 180 degrees, but the angles around a point will have a surplus or deficiency. The angles between two curves are also not invariant under choices of length (between which two points on the curve the third part of the triangle is drawn). All of these weirdnesses however should conceptually at least hold curvature information of the surface. I haven't been able to derive normal gaussian curvature using this for example, although it feels like it should be possible. Is this even possible/feasible to do? If so, what would it usually be called? Or is there some reason this approach would be worthless(other than being weird/unconventional)? Also, am I wrong to assume that this could then also be extended to sectional curvature information in higher-dimensional manifolds?