I know this may not be what you want to hear, OP, but has the family considered having this family member declared mentally incompetent so someone can gain power of attorney? With that, you’d (I think) have the legal power to remove animals and get others vet care.
As someone else mentioned, you should check the zoning laws in your area, and see what the limit is on the number of animals that can be on a certain amount of acreage.
I’ve seen enough hoarding shows to know that there is no way to reason with them, and even if they finally break down and say they understand - they don’t. They’re just buying time for you to turn your back so they can acquire more ___.
I really feel that it’s going to take the entire family descending en masse on this relative and forcibly taking the animals to get the results you want. Of course, they will go right out and find more, but that will be more controllable. As the animals come in, you rehome them. Will they be happy? No. Oh well.