In my opinion, one super important thing that isn't stressed enough is the definition of "furry" itself, being someone who is "interested in anthropomorphic animals". This means animals with any human characteristics, be that talking, walking, or just being more a human's level of intelligence; Mickey Mouse, Pikachu, and Jake the Dog are all examples of furries BY DEFINITION. If you are a human who is a big Lion King fan (interested in an anthropomorphic animal), then you are by definition a furry. This doesn't mean you must label yourself a furry, but the reason people shun the furry label anyways is just because they haven't checked the definition of "furry". More people are furries than they think, and that's fine. Humans have a history of thinking that animals with human characteristics are cool.
Even the traditional way people think of furries (animals shaped like humans) makes 100% sense. Dogs and cats evolved to look cute so humans would take care of them, and humans naturally are aesthetically attracted to the way that some animals look. A "furry" in the sense most people think of (again, bipedal plantigrade) is just a cute/aesthetically pleasing animal head slapped onto a human body. By that logic, even yiff makes sense- sexually attractive human body, aesthetically attractive animal head, fluff here and there, paws, and tail. I don't yiff, and much of the fandom doesn't, and that's obviously not the most important thing to focus on (I digressed a bit), but it does make sense.
In short: Being a furry does not mean I want to have sex with a cat. It also does not necessarily mean I like bipedal, plantigrade, anthropomorphically-shaped animals. I could just like Simba and still BY DEFINITION count as a furry. The real definition of what it is to be a furry needs to be more widely spread and the label needs to be less scary to people.