A simplistic view is that you need to add nourishment if you take nourishment off.
If all the muck from all the grazing animals goes back on the ground, things should be fine.
If a hay or silage crop is taken off and sold, there is a deficit which needs to be made up.
If a hay or silage crop is taken off and fed to your own animals, and the muck put back on the fields, things should be fine.
Sounds like perhaps the land has been cropped and not replenished, so you have a deficit to correct.
Chances are it is simply that - hay or silage made and sold, no muck or fertiliser brought back on.
So, to replace what has been lost, you need either FYM (farmyard manure) or artificial fertiliser. Or maybe things like seaweed, or even human slurry, but I’ve no experience with those to share.
FYM is the best for all the reasons FW outlines. But unless you know where it’s from and how the animals and land have been managed, there are risks. Disease, weed seeds, persistent weed killers, persistent wormers.
Manure from an organic farm should be fine - but since organics can’t use artificial fertilisers, it’s highly unusual for them to sell any of their FYM.
Personally, with permanent pastures which have been managed as grass for more than a couple of years, so have well-established root systems and a healthy soil structure, I would be perfectly happy to spread a light dose of artificial to redress the balance. Maybe not all at once - see what tonnes per acre is recommended for the shortfall you have, and if seems like a lot, spread half of it now and half next year. Or even 1/3 each year for three years.
If you spread a lot at once you might find the grass doesn’t make nice hay, so I’d err on the side of less at one time.
Ex-BH used to use a little artificial occasionally on his hay meadows, as well as using all his own FYM. He farmed 440 acres in all, and cropped perhaps 45 acres. About 2/3 of his cattle would be in over winter, making FYM, most of which went back on the hay meadows. However, he did make a surplus of hay to sell, so his system did need a little additional nutrient on the hay meadows (which were grazed after cropping and sometimes grazed lightly in spring too) from time to time. I seem to recall he used maybe 4T of artificial over perhaps 5 years. The meadows were all always full of a good variety of grasses and lots of wildflowers.