Sorry but I still don't understand what is the issue here (both for the bird model and the basic sphere sample). There have been no changes in the way we export normals for several years now so it may just be a misunderstanding of what you can expect is stored in the FBX file?
Given the response I would imagine it's very possible it's been wrong for several years. There appears to be something about Maya FBX export in contradiction to your statement:
The geometry/normals saved to the FBX files are taken before the deformation nodes (skinCluster) so we don't depend on time based modified values.
@ilan.keshet seems to have proven that with the sphere example. Indeed the normals, unlike geometry, seem to be baked to the position they are in at the time of export regardless of the animation. That's what I'm observing at least. Default FBX export options, no history.
I tend to import everything in its bind pose on the first frame. From my understanding, popular game engines write their own FBX importers. I didn't run into this until I had an animated file where the first frame was not in the bind pose.
If I export the same animated file, but right before I export I put the skeleton in bind pose, I get correct normals:

Per your statement I would expect this result no matter what position I have the joints in. (Again this is a single combined mesh so that really shows the deformation is being applied ONLY to the normals.)
Thanks for your responses.