Logically what is so wrong with making use of euthanized dogs and cats?

Why waste euthanized pets? Every year 10s of 1,000s of dogs and cats maybe slightly more cats get put down because they can’t find homes. Why let the death go in vain? The fur could be used to line coats for the poor and homeless. Many of the homeless are stuck outdoors nearly 22 hours out of the day and in winter in a lot of areas that lowers their survival rate quite a bit. Since we can’t build shelters for these people because we rather out government sink billions into unwinnable wars of power and oil, there is a resource that don’t cost much that can be used to help. Logically the meat should be used to, but I think many have too big of an emotional stumbling block to eat any meal in which dog or cat was an ingredient because they will feel guilty they are eating “Fido”, “Spot”, or “Barky”. All those doggie and kitty pelts are just going to waste when they logically could line some gloves or jackets for the homeless. How different is that from doctors descending like vultures over a fresh dead body to harvest whatever usable tissue they can to pass on to the living? Aside from the emotional repugnancy maybe many would have what would be reason logically not to make use of animals?