Why is the "topologist's sine curve" not locally connected?

The most likely reason is that it is less clear what happens in neighborhoods of $(0,0)$ compared to what happens in neighborhoods of $(0,y)$ for $y\neq 0$. The author is only trying to argue that the space as a whole is not locally connected so does not care whether or not the space is locally connected at $(0,0)$. The author likely felt (either correctly or incorrectly) that more of an argument would be required to demonstrate the lack of local connectivity at this point.

But, you can rest assured knowing that the space is in fact not locally connected at $(0,0)$ and the same argument can be applied here.

Draw the intersection of a (0,0)-centered ball with the set. How many "chunks" you see?