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.
Building structure requires serious listening, serious reflection, and serious imagination. All this requires experience, and no matter how experienced you are, it costs you. We spend our time and nerves to save users their time and nerves. Well-designed things give us the invaluable present of time. Well-designed products do not just save us time, they make us enjoy the time we spend with them. They make us feel that someone has been thinking about us, that a nice person took care of the little things for us. This is mainly why we perceive well-designed things as more beautiful the longer we use them, and the more used they become.
Design has very little to do with what tools or methodologies you use, or what your job title is, or what you have a degree in, or even anything like “creativity”; design is about your relationship to constraints. Rather: to what extent are you defining constraints rather than just obeying them? Design is about taking a universe of possibilities and converging onto exactly one outcome. Being handed a set of constraints which you treat like immutable laws of physics (because many of them are) and solving within that envelope is what engineering is. To wit: what most designers are doing most of the time is actually a form of engineering, and engineers are always doing at least some design.
This is because genuine design—the power to define constraints—is a privileged political position within an organization, and not everybody can occupy it. In other words, the “seat at the table” comes first. Design is Steve Jobs infamously dropping an iPod prototype into his fish tank, pointing at the bubbles coming out and yelling at his staff to make it thinner. It doesn’t matter what your title is; Jobs is the designer in that scenario.