To an extent, yes it would likely do that. Though on the other hand running into the maximum capacity limitations would not look pretty. Even countries that have a just bit higher grass-fed production than others have a fair number of issues (and still use plenty of supplemental grain)
For instance, in New Zealand, they use a massive amount of synthetic fertilizer on grasslands to try to make it keep up for dairy production
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.
theconversation.com/11-000-litres-of-water-to-mak…
Or in the UK and Ireland where grass-fed production leads to deforestation and they still need additional grain on top of it
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