Most rescues and shelters appear to take little to no action to test the cats they place in homes for various common diseases.
A fair number of my friends have complained to me over the years that they got cats from rescues, rehomers and shelters that either sickened or died, sickened or killed or at the least exposed the cats they already have to various diseases, etc.
It seems that cats brought into group housing very often pick something up.
My vet also told me that any time I pick up a cat that's been living wild, it has the potential to have been exposed to a large number of other cats and other diseases.
One friend of mine told me she no longer gets any cats from any rehomers or shelters of any type - she only buys cats from breeders that have wound up with some kittens that are not suitable for the show ring.
In some breeds, animals with certain faults would be humanely put down at or shortly after birth - she says. So she just talks with the breeders she knows, and say, if a kitten is born with the wrong color or such, she will take the kitten for a small payment.
She told me these cats are tested for more diseases, but I don't believe that all cat breeders are as diligent to deal with as the two or three ladies she knows so well and is familiar with.
I think the bottom line is that it is just a problem. I suppose when I get my new cat, I could pick it out and send it directly to the vet, have it tested for all dangerous diseases, and have it humanely euthenized at the vet's if it has a serious problem. But can most people actually afford to do that? What to do? Just hope you are lucky when you bring a new cat home?