Power
You really do hold all the cards here. First, your decision about an athlete's fitness to play affects their chance to rack up records on the field (or on the court, or in the pool...whatever). That means if you scrub them for medical reasons, even if only for a few days, that affects their chance to make the headlines.
Next, remember that if they can't toot their own horn, they can't help his team either. Therefore, the coach has to plug in a sub, which affects the team dynamics. Now we've got an injured athlete, a frustrated coach, and a team floundering around for chemistry when they were on their way to the top. Nobody's happy...except perhaps you, of course. You've hosted three press conferences this week, which has resulted in a few new patients, which was the aim of your power play all along.