This study examines AI adoption among Finnish healthcare SMEs through semi-structured interviews with six health-tech companies. We identify three AI engagement categories: AI-curious (exploring AI), AI-embracing (integrating AI), and AI-catering (providing AI solutions). Our proposed threefold model highlights key adoption barriers, including regulatory complexities, technical expertise gaps, and financial constraints. While SMEs recognize AI's potential, most remain in early adoption stages. We provide actionable recommendations to accelerate AI integration, focusing on regulatory reforms, talent development, and inter-company collaboration, offering valuable insights for healthcare organizations, policymakers, and researchers.