To use your example, your cousin was put in place due to nepotism. This is very much covered by my "some other ism" as corruption or at least bad behaviour. Cronyism and nepotism both bypass natural "best person for position based on merit" selection criteria.
Integrity is well within performance measures under a meritocratic system, eg police who are corrupt can't be seen as acting as the ideal for police officers or authority by any useful measure (merit). If you hired your cousin that nepotism impacts both yours and your cousins integrity and therefore merit. This bypasses the meritocracy and is actually simple corruption.
Plenty of other "isms" are also really just corruption in various forms: sexism, racism, tokenism, ageism. I'm sure there are plenty more.
You are talking about getting rid of ranking, possibly the concept of merit. It isn't possible to have a meritocracy with no concept of merit, but that is much stronger than not being a meritocracy.
For example, I could rank a team of 5 from A+ to F, then put cousin Ahmed in charge because he is a relation of mine. That would be a system that is not a meritocracy, because to be in charge a person would have to be related to me, but there is still a concept of merit. I would expect my cousin to then consider merit carefully as he divides up tasks.