That's a little simplistic that you can't spread manure on a working pasture. This guy says it's EXACTLY the right thing to do. He overstates the value of manure as fertilizer a bit though, by saying it's a substitute for chemical fertilizer, but that's just the common myth that we've already covered here.
One reason one might decide to use compost tea on your pasture instead is if the manure isn't really thoroughly composted. If you dump poorly composted manure on your pasture, you could to be dumping loads of weed seeds. I respect that concern. But nutritionally, compost tea is inferior to compost, which is already a somewhat inferior fertilizer. That's because you're taking all the nutrients in the compost, dissolving them in water, and spraying them on in a highly leachable form. Chemical fertilization is often done this way as well, in that chemical ferts are often highly water soluble., but they are vastly more powerful nutritionally, so you can afford to lose a lot to leaching.
So if weed seeds are a risk, then just properly compost your manure before spreading. Manure is rich in nitrogen and will naturally compost at a high temperature, killing everything, including weed seeds. You can say, oh, I'm too lazy to properly compost. But I'll set up a big plumbing outfit to make tea. That doesn't make a lot of sense.
What is the problem with livestock out walking on a thin layer of manure? Before you muck out their stalls, that's exactly what they're doing.
