Is the local bone rotation important if the anims play correctly?

I have exported an animated non-humanoid character from Blender to Unity with FBX 6.1

In Blender I recalculated the bone roll with global +Z axis

In Unity the animations play correctly in the preview window, also the whole character is aligned correctly (pointing in the direction of the red arrow in preview window).

When an animator controller is assigned the anims play correctly in the editor window and in play mode.

However: the local bone rotation of each bone is all over the place and doesn't match unitys Y up/Z forward. I tried nearly every possible combination of bone recalculation in blender but nothing helps.

So my question is: does the local bone rotation matter at all if the anims play correctly?

I'm thinking e.g. of a headlook controller, aiming with arms in the direction of the crosshair or some other fancy coding stuff (<- not a programer by the way, therefore I'm curious about that).