Grass will in time evolve into thickets, brushwood and to forest. Grazing (and hay) ensure that the pasture can continue. Manure provides food. So cattle can maintain the grasslands. And vice versa. In this symbiosis the grass feeds the cattle that eat other plants that take away sunlight and crowd grass.
Grazed pasture is so successful because grass holds its investment largely underground. It stores food supply in the roots. Growth occurs in a stem thickening (crown) just above or just below the ground. As a result, it matters little that the grazers devour wispy leaves above the ground. From the energy reserves in the roots stool soon new leaves. Also e.g. after burning.
By flourishing (generative reproduction by seed formation) aboveground shoots die off. Grass needs sufficient leaf area to store energy from sunlight into the roots. Which in turn provide back the short grass the energy to grow. A good pasture makes annually per hectare up to 4,500 kg root mass. The preservation of this circuit and the balance between above-and belowground biomass are required to maintain proper turf. Roots and leaves strive for balance. Frequent defoliation (by intensive grazing, mowing e.g. lawn) will kill many roots.
The wettest meadows were traditionally almost exclusively for hay, and therefore not trampled and compacted.
Some grasses like (tall) oat-grass (Arrhenatherum elatius) tolerate grazing. They are found in foods rich and moist roadsides that are mowed twice a year. When grazing oats disappear and the (crested) dog's tail (Cynosurus cristatus) that just needs grazing appears. Various types of grasslands are so named for their primary vegetation.
Etgras (translate as: day-grass?) is nutrient-rich fresh young grass that comes after mowing. After mowing a pasture can still be grazed.
Nowadays is machine mowed. That seems even to go (good) with blunt knives. The disadvantage is that the bristles are not cut, but very frayed. This gives the grass a regrowth delay.
There is always a spontaneous establishment and natural evolution of species. Although not always the way we would like it. I curse a few years at the Ivy in my pasture (probably because I let the grass devour too short by geese?).