Background/Objectives: The concept of educational systems thinking shows the rugby educational system as a network of multiple interactive and interconnected elements. However, the frameworks presented in the literature for developing sports practice through an educational vision and multisport engagement do not always have direct transferability to instructors’ training courses. The study aims to evaluate the correlation between rugby instructors’ methodological and didactical competencies and compare them with children’s motor competence and psychological aspects. Methods: Two hundred twenty children (9.6 ± 1.1 years) and 50 instructors (39.0 ± 13.6 years) were enrolled. Instructors’ competencies were investigated during the internship through evaluation sheets, questionnaires, and video analysis. Children’s motor competence and psychological aspects were investigated through motor tests, questionnaires, and video analysis. Results: The correlation between methodological competencies and didactical-communicative competencies showed a positive significance, especially regarding production styles. Motor competence and play skills in children showed poor results. The group of instructors with more methodological competencies (a greater number of teaching styles used) showed a strong correlation between motor competence and psychological aspects. Conclusions: For successful teaching, knowledge of specific technical skills is not enough, and it is important to consider all the factors (in particular, the teaching-communicative and methodological skills) that contribute most to instructors’ skills.