First. You are totaly correct in that Assault support has nothing to do with Fatigue. My mistake, dont know what I was thinking of, sorry.
Unit can have multiple states independent of other states. Squads can be Fresh, Non-Fatigued, Fatigued, in OP-fire, and/or pinned/disrupted, Non-Pinned. I suppose they can also be Non-Fresh.
My take on the assault rules, is still that they were graftet onto the rest of the system, instead of having been incorporated. Thus creating multiple exceptions to normal gameflow.