Your cousin's case is a legitimate medical reason for an exemption and is why the rest of us need to maintain herd immunity by getting vaccinated. A bunch of guys I work with were refusing to get vaccinated just because they didn't want to be told what to do. Guys like that are actively endangering the people who CAN'T get vaccinated, like your cousin. Herd immunity is critical in order to limit the spread of diseases and protect the people who can't get vaccinated or who have other medical complications that put them at high risk.
Legitimate medical exemptions are fine. Deciding that you "just don't want to" should come with consequences. My partner's 87 year old mother could be killed by Covid. We have all maintained flu/Covid shots to protect her.