I agree that the paper's title is somewhat misleading, since you still do need to assume an invariant speed to rule out the Galilean transformations.
However, this derivation does greatly narrow things down before the invariant speed comes in: at the point where the invariant speed is assumed, you already know that there are only two alternatives: an invariant speed (Lorentz transformations) or Galilean transformations. So it's much easier to see why you would assume an invariant speed; the assumption isn't just pulled out of thin air at the start, it is seen to be one of only two alternatives that are compatible with the principle of relativity.
I agree that the paper's title is somewhat misleading, since you still do need to assume an invariant speed to rule out the Galilean transformations.
However, this derivation does greatly narrow things down before the invariant speed comes in: at the point where the invariant speed is assumed, you already know that there are only two alternatives: an invariant speed (Lorentz transformations) or Galilean transformations. So it's much easier to see why you would assume an invariant speed; the assumption isn't just pulled out of thin air at the start, it is seen to be one of only two alternatives that are compatible with the principle of relativity.