I learned to cut dovetails by hand a year and a half ago. It seemed daunting, but with perseverance and cutting many ugly practice pieces with inexpensive wood, I managed to do well enough to sell several boxes and urns with dovetail joinery.
Router cut dovetails have their place. If a professional woodworker is consigned a dresser full of drawers in which the customer wants dovetailed drawers, it only makes sense to use a router to be efficient.
My current work is comprised of making jewelry boxes and cases, cremation urns, and boxes to fit essential oils. Cutting dovetails for these takes longer than by using a machine, but I have found that the process is meditative, that I feel harmonious when sawing dovetails. The tranquil quality of the time is invaluable to me as a woodworker. It is empowering to cut dovetails by hand, the only way it was done many years ago.
A dovetailed box or drawer is strong. Often the drawer front is dovetailed and the back is rabbeted. Because there is more force applied to a drawer front, dovetails are commonly used.