I think it's a bit silly how you can only get base quality milk and meat if you raise your cattle on a pasture. The Swiss drive their cows high into the mountains so they can feast on the lushest grass and give the best milk for fine cheeses. Cowboys drove cattle all over central USA, chasing the best grass that grew after rain, so that the beef reaches it's highest deliciousness potential. Fact is, free range animal products tend to taste better, and through-feeding cattle in a small enclosure is only done IRL because it's much cheaper.
Anyway, I'd like it to be viable in H&H to do that. The grass should deplete, of course, but instead of just disappearing like it does now it ought to lose quality. Quality of a grassland would be raised periodically on a local level, even if it doesn't make much sense without weather and seasons. That is, if you want the very best feed for your cattle, you'll have to follow the good feeding instead of just keeping them on one big pasture all the time. The grass will slowly diminish in quality on it's own (represents wildlife eating it), so you need to be at the right place at the right time. You can also use a scythe to cut it and get hay, if you find it easier to leave the cattle behind and travel on your own (though it means more work).
The real purpose here is to make nomadic life a viable alternative to the sedentary one, as a nomad could support a much larger number of animals. Also, this would lead to cattle getting lost now and then, which would give the newbs a chance to catch some pre-domesticated cattle. Who knows, if enough escape they might even establish a feral cow population (second generation and beyond), which could be a bit easier to tame than aurochsen.