In my honest opinion there are a few mistakes that have been done, but nothing that cannot be remedied.
You never mentioned where you are from, but I have had really good luck with a pasture forage mix from
Tractor Supply that was decent in price. It was $50 for 25 pounds, at a seeding rate of 25 pounds to the acre. I do not have goats, but do have sheep and even in the few times I have overgrazed it, it has done really well. It also did well during (2) severe droughts. That being said, what I planted was the "Northern" mixture as I live in Maine, but for those further south, they have a Southern mixture, and I would recommend that obviously. It has a really nice mix of cool and warm season grasses that fatten my sheep up nicely.
As for the fertilizer, I am not a huge fan of commercial fertilizer as I prefer manure, but have used it some because it was required by a grant. In your case though you used the wrong mixture because 10/10/10 is a general fertilizer and not a great one for starting grass. Again I am not a fan of commercial fertilizers, but even if forced too, a person would want to use something like 05/13/41...basically anything low on nitrogen. All that will do for beginning fields and pastures is give nitrogen for the weeds to grow, which will stunt the grass that you are trying to grow. What a person is really looking for is the middle number, that is what gets the roots established. BUT this is where a person has to be really careful because nitrogen (the first number on the NPK) dissipates very quickly, and while the last number, Potash is sucked up by the plants, is just just not released if it is not used...but phosphorous...that is the bad guy. That is what causes the pollution if it is overdone. Really try and never go over on the middle number no matter if manure or commercial fertilizer is used; it is the bad guy of the NPK world.
Now with all that said, most likely manure will be a far better alternative. 90% of the
land out there needs more organic matter, and this helps increase that, along with getting the NPK levels up. Ruminant poo 85% of what they eat back out, so really you just have to do a manure test, see what they are putting down into the soil, and then supplement the 15% that is being grazed off and not returned as poo. In my case, my pastures need 50 pounds per acre of 10/10/10...but keep in mind, my fields are established. I dislike commercial fertilizers so instead I use manure gathered over the winter months to get me where I need to be. At the same time, a ruminant does not kill off a lot of seeds in the sward they eat, so spreading manure often means overseeding too.
It is possible to get certain grasses to grow that lessen your need for even manure, but sadly we do not live in a perfect world. A lot of the grass types that allow for nitrogen fixation, do not tolerate grazing so well. In my hay fields I use a variety of grass types, but have lately switched to a 50/50 timothy-clover mixture, I need almost nothing for manure for that, but that would
not be a good grazing sward however.
This is about the best I can do for a reply as really there is not
enough information given to help you any better than this.