The impact animals have on grass is tremendous. They crop it short, trample it into the ground, and shit on it. All of which, when not done excessively, is exactly what the grass needs. Grazing stimulates new growth. Stomping and scratching "till" plant matter into the soil without disrupting it. Manure adds essential nutrients back to the ground.
A key to this process is the word rotational. Keeping the animals in one spot and never moving them will eventually result in no plant matter, and if you don't add enough bedding or remove manure on time, an unpleasant smell. You need to put the animals into a pen or "tractor" and move them every day or every few days, allowing them to graze and forage without eating and digging everything up. Each patch of grass needs a rest period where no animals are present for a while so the grass can regrow and the manure break down into the soil. This is a well documented, low impact, and highly successful process that many small (and some large) farms and backyard livestock enthusiasts employ.
You cannot accurately replicate this process with mechanical and synthetic means. You create a sterile polluting environment devoid of important microbes, insects, topsoil, and plant diversity. It needs constant attention because it's not a thriving symbiotic ecosystem.
A huge benefit to small scale livestock such as chickens and rabbits is they can give you that beautiful grass with very little cost. Chickens can eat leftovers, taking care of an important source of food waste. They also eat bugs and greatly reduce the umber of pest species without killing all the beneficial pollinators as well. Chickens provide eggs and meat in return. Rabbits love plants like clover and dandelions, put them in areas with weeds and they will take care of them for you. A few rabbits can provide several pounds of highly nutritious meat.
