I am a horse owner and pasture maintainer and have some insight, of experience, to share with you.
(1) This thing about privately owned animals sharing a commonly owned grazing area isn't just a medieval thing. It's current, and it is real. Four of my friends who have horses, and do share the dubious advantage of rent free, land, owned by an absentee owner. Their pasture is the most overgrazed, and unproductive, I have ever seen, the only visible green growth, being noxious weeds. The situation is truly hopeless, because each owner visits at a different time, and each has incentive to turn out her horse when she visits, to find any grass or palatable weed seedling which may have sprouted, at the expense of the overall herd. The incentive of each individually, has worked to be the doom of the collective, just as in the story.
(2) Sheep are notorious for selectively short cropping the tenderest grass shoots.
(3) While you are right that the capacity of a pasture cannot be known in advance, due to weather variability, the capacity tipping point is much more precarious than just having a slightly less productive pasture next year. What happens is this: Animals continuously seek the newest growth. As grass becomes even slightly less lush, the animals have to slightly increase the volume of forage consumed, due to the slight decrease of forage quality. Simultaneously, the grass begins to fall behind in its race to match consumption with growth. The result is that, passed its carrying capacity, the pasture growth deficit increases exponentially. A pasture that is imperceptively overgrazed in June, can be so overgrazed in July that it would need a couple of months without any grazing whatsoever to recover. It will still have a full covering of grass, the grass will just be a little too short, and beginning to rapidly fall behind in its fight for survival. If it is still grazed for just a few more weeks, now even by just a single horse, the pasture will be effectively destroyed, and need to be reseeded, requiring a full year without grazing to establish the new grass seedlings.
(4) It is very difficult for the animal owners of an overgrazed commons to rest the pasture, to effect a recovery, because the incentive is for each owner to "cheat", and allow her horse to graze for an hour while the other owners are away. In the case of my four friends, they have not even tried to make a pasture renewing consensus. (After all, it is not their pasture.) Nowadays, the effect of overgrazing is just that the owner will have to buy more hay for the year, and the animal will enjoy less fresh grass. But, in medieval society, presumably with little more hay available than what was needed to overwinter that portion of the herd not butchered, an imperceptibly overgrazed commons in the spring could have required a tragic herd reduction, by mid summer. There is a strong amplifier effect. One too many lambs in April, give one too many sheep in June, which might mean twenty sheep have to be culled prematurely in July, else by August, hundreds would need to be butchered, or herded to neighboring communities, either of which would be at a desperation price. Tragic indeed in a primitive agrarian society, One too many in spring, if not culled early, could jeopardize all in the coming winter.
The rate at which such a system collapses can vary enormously, and accelerating failure is quite common. When it comes to pasture land, I am not surprised to learn that a small pasture would fail more rapidly than hundreds of square miles of BLM land in NW Nevada (where this kind of thing is also going on right now) or the entire southern periphery of the Sahara Desert.