See Chainmail
Page 8. They are a bit more complicated with charging rules, but you can use fatigue out of combat too.
So my edition of Chainmail has these fatigue rules on Page 11 ... for mass combat. They are not found in the man to man rules and no reference is made to them in the D&D alternate combat system or any subsequent edition. I would note that the effects are profoundly different in the chainmail mass combat system. This rule effects morale (super important here and meaning that combat is unlikely to last 3 melee rounds), is non-cumulative, and that it doesn't give you -1 to attack but rather reduces the quality/class of the troops to one lower.
In chainmail I think it would work like this (basically):
10 light foot are fighting 10 heavy foot:
Light foot roll 5d6 dice and only kill 1 heavy foot on every 6.
Heavy foot roll 10d6 dice and kill 1 light foot on every 5 or 6.
If the Heavy Foot are exhausted they will fight as light foot so:
Light Foot roll 10d6 and kill 1 heavy foot on every 6
Heavy Foot roll 10d6 and kill 1 light foot on 6
At the end of the melee round we'd check morale but i won't get into that.
It seems rather distinct from D&D's methods - though of course in OD&D you can always adapt Chainmail, and while I think you are making a cool point and performing exactly the kind of laudable OD&D hermeneutics I love, it doesn't strike me as nearly as simple as a "rules not ported to AD&D" situation. It's an interesting concept and idea, but it would need significant adaption, even to Chainmail based OD&D. It's akin to saying that the fatigue and starvation rules of Wilderness Survival are part of OD&D - at least to me.