Then you are obviously not running an intensive NZ grazing system, which was my original point! NZ system is intensive grazing/over stocking with little regard to the environment or animal welfare. Plenty of great examples of milk from grass in Cornwall, but not the NZ way. Those that practice this are now in the sh!t, no soil structure, no grass, no alternative feed and as farmers that choose to make profit from stock then there is a duty of care to look after animal welfare, for example provide feed!You are right about cows producing carbon on forage but not total carbon emissions.
Growing grass captures carbon through photosynthesis. This ends up in the soil as organic matter (either through root growth or cows eating and shitting.
Moving soil, or bare soil produces carbon - burning diesel and laying concrete also do.
Bare soil over ay sort of wet period will produce run off - a poorly tillered arable crop will do so (although a slightly poached but not destroyed soil from grazing brassicas won't as much, oddly.) so growing anything other than a perennial crop is potentially producing carbon - before you play with the soil, this is made worse by the fact that cows shitting on concrete is actually more harmful in terms of emissions than on soil, where microbial action is much quicker.
Cows are healthier fed grass, rather than high loadings of grain which is effectively what whole crop and maize are. They are not well suited to walking on hard surfaces either, but they are quite capable of weathering bad weather outside.
I house cows for some periods but only because i do not have suitable farm infrastructure to access some areas and some of the farm is heavy clay in places. I can graze a fair proportion of the farm thought winter and didn't house any cattle in 2016/17, grazing right through. A university is measuring water quality of the river going in and out of the farm and the water is actually cleaner leaving than entering. This is in part due to a fair amount of maize being grown further up the catchment, something which the EA is less than happy about, especially when they umbilical slurry on it on the 1st of Feb, rain or not.
NZ is having problems in part because they have lost their way with high payouts. The systems being run in the 90s were much more interesting and progressive IMO