You really cannot talk about this in statements like they apply everywhere. Here's an example: around where I live, the number 1 activity by nature preservation groups when it comes to restoring grasslands of all kinds, is mowing. For a couple of reasons, but the main one being that nitrogen deposition fertilizes the soil too much, so it gets countered by removing biomass via mowing. Not doing that results is fast growers. Not because the'yre indicating a recovery process, just because they outgrow what was there originally. Which is the opposite of recovery.
leading to a much more healthy ecosystem than when you mow it all down to hell.
When I say "it's the amount, how and when" that implicates 'not mowing it all down to hell' is not the right way :)
Where I live the native prairie was naturally mowed by the bison roaming around, and so not mowing leads to a number of problems as the plants around here depend on that regular mowing. Various lightening strikes and the like ("Indians") caused regular fires as well, which took care of overgrowth. Mowing simulates that without the fire. (this is both good and bad - it makes for cleaner air, but those fires also put charcoal into the ground and so made the prairie net carbon negative over the long term which is something we desperately need)
That lacks an important nuance: in most circumstances (regarding soil type/nitrogen deposition), not mowing at all would lead to those patches being overgrown by fast growers leading to less plant diversity. Can happen in as little as a year or 2. So the mowing itself is not the problem, it's the amount, how and when.