I think 12 is a very good innings for a goat that spend most of her life in a commercial setting...., and you have probably made up your mind anyway. I will have any goat that is past breeding pts before the next breeding season, just to be on the safe side. Normally my old girls would have a retirement summer, I often graze an oldie with the youngstock, who will accept her as their leader and she won't have to fight with the fitter younger females.
My vet indeed puts my goats to sleep, I normally take them down to them in the van and it is done very quickly and painless in the van. Yes the same stuff as for horses, and I bring my girls myself to the knacker yard (not necessarily for the faint hearted but an important part of keeping livestock) on the way home from the vet. Very lucky as it's all within 5miles radius from the holding.
The knackerman (I have yet to meet a woman doing this job...) will also come to your farm and shoot the animal, I have found them quick, efficient and business-like if needed with my sheep.
I personally will not send any deadstock to the local hunt's kennels, but that is another option.