Vegan

A few things to consider. A minority of cows are free range grass fed, a very small minority are 100% grass fed and free range to an extent that matters, even in the case of 100% grass fed there might be fertilizers used. And make no mistake, cows on open pastures are not a good way to feed the human population from an environmental stand point, probably worse than factory farms honestly. Fields are terrible at sequestering carbon and lack a lot of the biodiversity of the forests most of them historically replaced.

In New Zealand HUGE amounts of fertiliser are used to boost the amount of grass grown and thus cows per unit area. Some ends in groundwater and a lot can end up in streams and rivers too.

Wouldn’t excrement be too valuable as a fertilizer and for producing methane power to just let it all run off into streams etc.?

A lot of it is sold off but the cows aren't always spending time where it can be collected. When they are out in the pasture the crap just stays where it falls and it will inevitably get picked up by rainwater.

I also wouldn't be surprised if this is including the manure that is being used as fertilizer as well. Because when it rains that manure isn't just going to stay on the field. There's one road in my area where, whenever it rains, you can watch a crap river flow down the road because of the sheer amount of manure running off the nearby field.

Cows are given antibiotics to prevent diseases (and to promote muscle growth, although doing so is now banned in the EU and China). They shit out a large fraction of the antibiotics, and these kill soil microbes (and make the survivors antibiotic-resistant, which is absolutely great news for public health). Without soil microbes to process them, the organic matter in litter does not stay in soil and instead gets respired into the atmosphere (accelerating global warming) or washed into lakes and rivers (causing eutrophication).