It increases methane emissions and doesn’t scale

We model a nationwide transition [in the US] from grain- to grass-finishing systems using demographics of present-day beef cattle. In order to produce the same quantity of beef as the present-day system, we find that a nationwide shift to exclusively grass-fed beef would require increasing the national cattle herd from 77 to 100 million cattle, an increase of 30%. We also find that the current pastureland grass resource can support only 27% of the current beef supply (27 million cattle), an amount 30% smaller than prior estimates

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If beef consumption is not reduced and is instead satisfied by greater imports of grass-fed beef, a switch to purely grass-fed systems would likely result in higher environmental costs, including higher overall methane emissions. Thus, only reductions in beef consumption can guarantee reductions in the environmental impact of US food systems.

https://iopscience.iop.org/article/10.1088/1748-9326/aad401

Taken together, an exclusively grass-fed beef cattle herd would raise the United States’ total methane emissions by approximately 8%.

https://iopscience.iop.org/article/10.1088/1748-9326/aad401/pdf

Further, plenty of the land that grazing takes place on is not naturally grassland, and the “grass-fed” that you’ll see anywhere are still getting grain as well

Most of the UK and Ireland’s grass-fed cows and sheep are on land that might otherwise be temperate rainforest – arable crops tend to prefer drier conditions. However, even if there were no livestock grazing in the rainforest zone – and these areas were threatened by other crops instead – livestock would still pose an indirect threat due to their huge land footprint

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Furthermore, most British grass-fed cows are still fed crops on top of their staple grass

https://theconversation.com/livestock-grazing-is-preventing-the-return-of-rainforests-to-the-uk-and-ireland-198014

Places that have tried to scale grass-fed production up have all kinds of problems. For instance, New Zealand often likes to tout its grass-fed production, but the production levels are so high that it’s a heavy polluter. It would require a 12-fold reduction in size in one region to meet the bare minimum standards for drinking water safety

The large footprint for milk in Canterbury indicates just how far the capacity of the environment has been overshot. To maintain that level of production and have healthy water would require either 12 times more rainfall in the region or a 12-fold reduction in cows.

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The “grass-fed” marketing line overlooks the huge amounts of fossil-fuel-derived fertiliser used to make the extra grass that supports New Zealand’s very high animal stock rates.

https://theconversation.com/11-000-litres-of-water-to-make-one-litre-of-milk-new-questions-about-the-freshwater-impact-of-nz-dairy-farming-183806