Pastures are a complex system of living and non living factors and you are trying to ‘use’ this system to your benefit. I too rely on pasture for my horses and am constantly trying to keep balance while my horses compact, nibble through, and generally tear up what can be a lush garden. A pasture will never be as neat as a farmers field but it can be helped so that your grazers have the benefit of a safe and comfortable place to live.
Some experts will suggest a heavy herbicide/fertilizer approach to pasturing. I would argue that there is nothing great about this approach. Certain unwanted plants can be overcome by planting competing plants or by wisely mowing. My pasture management style keeps in mind the idea that any place will have a natural ‘succession’ after change.
This term ‘succession’ I learned from a biology book decades ago. Say a volcano erupts and lava covers what use to be a forest. The damage is complete and nothing survives. Give enough time and this area will be a forest again. This succession relies on wind, sun, and really tough species who initially move in.
Your pasture use to be something different than a gated community for horses. It is trying to get back to that state. If you have a problem with a certain weed it is probably because there is not a crop that you wish there. If you can fill the niche that your weed is in, then there will be competition for the niche and weed may just get ‘weeded out’.
My ongoing problem is with gypsum weed. It has nasty thorny leaves and reproduces a lot. I have learned that mowing it only makes it more powerful but that harvesting the seed pods before they erupt will end the problem.
Of course there are neighboring farmers who do not take such precautions and let their water ways overgrow with the stuff. I blame the birds for flying over and broadcasting the seeds into my pasture so every year I have a few plants to watch and collect seed pods from. Luckily, these plants are annuals and if they do not set seeds, they are dead dead. A note about gypsum weed is that you have to incinerate the seed pods or they will grow wherever you throw them. I would think that you could put them in the municipal garbage if you want.
There is a better way to battle pasture weeds than to apply herbicides. For bull thistle, I walk the pasture with a spade and simply cut them off at the base. If this is done before they set seed, the job is done and is only a slight maintenance issue each year. Maybe my most important pasture advise is to spend time in it. If you are aware of what plants are where, then you can make decisions about how or when to manage them. And it keeps the horses friendly.