At the same time few things are as suspicious as an antipathy to the competent being in charge, especially when it comes from the existing elites. It just screams "How dare you interfere with my ordination by connections by being better!".
There are essentially multiple levels of meritocracy. A level 1 meritocracy would judge only by current skills - for better or worse. This may be less than impressive even if technically a meritocracy. It may say, result in most knights coming from noble families trained from birth, but exceptional individuals would not be barred just from their background. Strictly better than a hard caste system but not something to brag about. A level 2 would try to ensure some degree of access to skills and education to all and be more meritocratic. Public education of unequal qualtity would qualify. A theoretical level N would involve completely equal starting points and would thus have pure 'merit' as the decider, even if it only accumulated from luck and the normal curve. Which highlights another issue - the distribution of quality is never perfectly even, it tends to follow a normal curve of some sort.
As for 'solving' the issue. Ability begets ability - this is called education and practice and I doubt there is a true alternative. We would call it rightfully barking mad to ban education for the sake of equity despite education contributing greatly to disparate outcomes. I think that is one of those imperfections of the universe we must accept for now.