Handwriting slowly became a form of self-expression when it ceased to be the primary mode of written communication. When a new writing technology develops, we tend to romanticize the older one. The supplanted technology is vaunted as more authentic because it is no longer ubiquitous or official. Thus for monks, print was capricious and script reliable. So too today: Conventional wisdom holds that computers are devoid of emotion and personality, and handwriting is the province of intimacy, originality and authenticity.
I always took great pride in my handwriting, even many years before I started using a computer regularly. I still regularly write longhand in my journals and other offline writing. I don’t think everyone still has to write that way, or learn cursive, though. In fact, I kind of like the idea of it becoming a more esoteric art form.