FBX key is not generated when:
* You only select Motion to export.
* If you check Delete Hidden Faces
* If you select any option in Include Motion section other than Calibration AND the Universal T-pose Editing is NOT checked.
Note: There is a bug. If you select Current Pose, the Universal T-pose Editing is unavailable to select.
So if you need to export as Current Pose, you need to select Custom first, then check Universal T-pose Editing, then switch back to Current Pose and export. Only then the key is generated.
In this case, the T-pose is still at the first frame and then there is a short clip transforming a T-pose to *Current Pose* (the one you have in CC viewport).
The above limitations in key generation somewhat makes sense.
Because even if you had the key, the model will not import correctly back to CC, if it did not have a T-pose as a first frame in animation/pose for both Pose Position and Rest Position (Blender terms in this case).
I figured I'll show you what would happened with import even if you had FBX key, but failed to export in T-pose from 3-rd party app. This explains in part FBX key availability during export from CC as well.