Unicorns certainly existed. That’s why cultures with no contact depicted the same animal.
Not only do they exist, but you can see them at most major zoos. You have to keep in mind two things:
1) The same people who described them as a horse with a horn also described a hippo as a “river horse.”
2) Unicorns were said to have massive, thick legs and feet like an elephant, super-tough hides, and were extremely rugged.
3) The horn grew out of its nose, not its forehead.
Now, what creature looks no more unlike a horse than a hippo, has a massive horn on its nose, has legs and feet like an elephant, is built extremely ruggedly, and has a thick, thick hide?
The fairy-tale girly horses are the imagination of someone who read no further than “horse” and “horn.”