There’s no waiting room at Linnea Meyer’s tiny primary-care practice in downtown Boston. That’s because there’s rarely a wait to see her. She has only 50 patients to date and often interacts with them by text, phone or email. There’s no office staff because Dr. Meyer doesn’t charge for visits or file insurance claims. Patients pay her a monthly fee—$25 to $125, depending on age—which covers all the primary care they need.

“Getting that third-party payer out of the room frees me up to focus on patient care,” says Dr. Meyer,...