What's the etymology of 'of' after verbs?

Your question seems to be based on the assumption that "because grammatical (eg: genitive) cases affect nouns (and not verbs)". But that is not the case (sorry).

A case is generally expressed through morpheme marking on the noun. But the specific case assigned to the noun depends on entirely on the head of the phrase which could be a preposition (e.g I showed up with her.) So when the case marking disappeared, the analytic replacement of the synthetic genitive simply took on the role.

In modern grammars and dictionaries of English, the 'of' is included as part of the phrase which simply takes the noun as an argument. But you could just as well think of the 'of' as part of a 'marking' on the following noun phrase that fits into the argument requiring a genitive.

There are a number of metaphors used in linguistics to talk about this. The metaphor from logic is one of argument taken by the verb. Generative grammar (most commonly in its GB incarnation) talks about thematic roles or theta roles. But there's also a tradition of European syntax that talks about the valence of the verb which draws on the idea of valence from chemistry.