95% of the information on the web is written language. It is only logical to say that a web designer should get good training in the main discipline of shaping written information, in other words: Typography.
Typography has one plain duty before it and that is to convey information in writing. No argument or consideration can absolve typography from this duty. A printed work which cannot be read becomes a product without purpose.
Information design is not about the use of good typefaces, it is about the use of good typography. Which is a huge difference. Anyone can use typefaces, some can choose good typefaces, but only few master typography.