Themes
Measured, not named
A projection is usually called conformal because that is its name. Here it is called conformal only after the angular deformation has been computed at several hundred points and found to be zero.
A theorem, not a limitation
No map is faithful, and this is not a shortcoming awaiting a cleverer cartographer. Gaussian curvature is intrinsic, a sphere has some and a plane has none, and that settles it.
What survives a change of coordinates
Scale along the meridian and scale along the parallel depend on how the sphere was parameterised. The principal scales, the areal factor and the angular deformation do not, and only those are properties of the map itself.
Every figure here is a projection
The page is flat, so there is no way to show the sphere except by projecting it. Even the globes are projections with their own distortion, and no picture here is the undistorted truth.
The trade-off is forced
Conformal means the principal scales are equal; equal-area means their product is one. Both at once forces them both to one, which is an isometry, which cannot exist. The trade-off is two lines of algebra.
Computed, not quoted
Scale factors, areal ratios, angular deformations, geodesic distances and the areas of countries are all computed from the projection's own formulae. None is a number recalled from a table.
Purpose before property
Asking which projection is best is asking an incomplete question. Every one of them minimises something, and the only useful comparison is between a projection and the job it was chosen for.