This is misleading. With a one-way function, we are interested in making it hard to find any preimage, not just the one you originally thought of. There are lots of functions whose range is small, but where it's easy to find a preimage (eg: take the first 160 bits of input; XOR all 160-bit chunks together; always return 0).

The avalanche effect (referred to in the last sentence) is important as a heuristic. Among other things, it makes it harder to go from an "approximate" preimage to an exact one. If we had f(x) = y, where y is similar to our target y', we shouldn't be able to find x' with f(x') = y' just by looking at the neighborhood of x. But this doesn't rule out more "clever" ways of tweaking x, and it doesn't obviously stop an attacker from deducing some property of x'. So for one-way functions, what we really want to assert is the nonexistence of an algorithm for finding (properties of) preimages, that is any better than just trying lots of new values of x.