RE: "Is self-care a helpful concept for Lutherans?"

For this Lutheran, it most certainly is. The term succinctly encompasses a variety of activities that support God-pleasing stewardship of body and mind. Such self-care benefits not only oneself but also family, friends, colleagues, and anyone else with whom one may interact.

RE: "Self-care gurus will teach you that in order to take care of yourself, you need regular breaks from whatever in your life is hard or unpleasant."

As a general rule, taking regular breaks from difficult or unpleasant tasks does seem wise. As Jesus once said to his disciples, "Come with me by yourselves to a quiet place and get some rest" (Mark 6:31 NIV).