I was never a huge fan of FE5's fatigue mechanic anyway, not because it was too black-and-white (that is a problem to less experienced players), but because it was something that I never cared about 90% of the time but then had to meticulously plan around 10% of the time. I think one big problem with the fatigue system is that it's really, really hard to plan around:
First of all, fatigue state is compared to max HP, and max HP changes over time, so it's harder to predict when a unit will cross a fatigue threshold.
Secondly, even someone such as myself who plans an entire playthrough ahead of time can't accurately plan how much fatigue a unit will accumulate by a certain point. So fatigue is more of a tactical thing, i.e., you try to play around it as you go. I can't really give an objective argument against this; I can only say that it's unsatisfying to deal with something that's unpredictable and restrictive.
Those are my grievances with the mechanic as a broad concept. As for your suggestions, in addition to the complexity (e.g., why make moving 5 tiles accumulate fatigue? It's way simpler to do all or nothing movement = +1 fatigue), I can see there potential for abuse in the mechanic of a fatigued unit retreating at half HP, because that essentially means that unit is in some sort of Casual Mode state for that chapter.
That's a bit different from managing fatigue. Managing fatigue is more like managing weapon/item uses, but it's also much more feasible to manage weapon uses because fewer actions cost weapon uses.
Honestly I think that fatigue would work much better as a concept if it worked in inverse. A fatigued unit, for example, might never get weapon triangle advantage, and that's the worst thing that can happen. A fresh unit (i.e., one that has sat out a chapter and starts battle at 0 fatigue) might get some sort of non-trivial bonus for the entirety of that chapter. It seems better to encourage unit rotation by rewarding the player for letting a unit sit out than by penalizing him for not letting a unit sit out.