I get that and I suppose our obsession with Rye is the problem as it can't stand up on it's own. But I still think that most farmers do not give grass adequate rest periods and certainly don't let it grow long enough. The root length of a grass plant usually mirrors the height above ground. So, if you constantly graze your grass right down, the roots never get deep. This aids compaction and also reduces the drainage potential of the ground. Letting your grass grow higher lets those roots go deeper and you still have a lot of leaf on the top. OK, letting it go too far is not a good thing. Grazing hard in taller grass does 3 things. It provides nutrition for the animal from the leaf of the grass. The stems are trampled and mulched back into the soil giving you plenty of humus as you say and all of this means you end up with a shorter plant which means those long roots die back to match and provide a huge amount of composting OM deep in the soil.
I suppose we are talking about extremes and I take your points. You can't just let it go. But I am convinced that most farmers graze too soon, not hard enough and don't leave enough rest. Better to get to that sweet spot and hit the grass hard with lots of animals for a short time and then be gone.