[Meritocracy] is not something that only occurs in the top branches of the government. It’s something that should occur in every level of every organization, in every office and in every pay-grade.

Please look into Feudalism. Then please look into why it has faded into obscurity. The Japanese had a particularly poignant understanding of it.

The idea is that you get enough rewards (money, social capital, etc.) that you will find the work worthwhile.

This is capitalism or social credit.

“who’s on top”, but in a perfect world no one is.

Popularity contests are a bad way of making choices

This is anarchism. Which leads to mob rule, the definition of power in the majority, and then to fragmented autocracies. ie Individuals grouping up to gain advantages then forming gangs, tribes, and engaging power struggles.

Right we do the worst option, in which the governing body distribute funds based on political power.

Which country is "we"?

The law is a bunch of rules, chosen by people in power

Not laws, 'the law'. As in the determiner of how the rules apply to the people. This is typically the police, legal interpreters, courts, on up until you hit judges and legislators, who hold the power to modify laws.

We don’t have to have the perfect definition

Because perfection is an illusion. The reason behind outdated-laws, governments struggling with complexity, and loopholes is precisely because any time there is ambiguity, there exists abuse. Meritocracy being founded on an ideal implementation where everyone in society supports the idea and nobody tries to abuse the system is folly, bound to fail at first brush with ambiguity.

There are many possible algorithms.

Forgive my bluntness, but your ideals are half baked, complexities waved away as if the pieces will fall into place after taking the leap, and tried but not studied. You would need a much better understanding of history and the governments that have already existed before you could convince me meritocracy can survive beyond dreams and ideals.

Apologies. I wish you luck on your journey through life.