You ask an interesting question.
The conventional rule is that adjectives qualify nouns, while adverbs modify verbs. You are right to feel uncomfortable about
He is arguably the best actor of his generation.
As Greenbaum puts it in The Oxford English Grammar [4.26 page141]:-
Adverbs are a heterogeneous class, varying greatly in their functional and positional ranges.
He also points out that some adverbs (or rather, adverbials) modify not a verb or an adjective, but a clause or sentence as a whole. he distinguishes three types: conjuncts, disjuncts and adjuncts. As far as I can tell from his examples, your “arguably” would count as a disjunct adverbial. This type, is described as follows. He says:
Disjuncts provide comments on the unit in which they stand.
And then on p.147, having distinguished style disjuncts (such as “And the second uh purpose is in fact involved in sex or more strictly I suppose the exchange of DNA”) from content disjuncts, which he explains as follows.
content disjuncts may be modal (commenting on the truth-value) ...[he gives as example: “This is probably a woman’s size”]... or evaluative (making a value judgement) ... [he gives among his examples “Moreover, Irish voters have wisely never given him an overall parliamentary majority”].
Greenbaum includes arguably in his list of the modal kind of content disjunct. Here is part of his list
Admittedly, certainly, clearly, ...undoubtedly, apparently, arguably, ... possibly, ... theoretically.
Greenbaum is arguably correct. Or I could have written that “Arguably, Greenbaum is correct.”. Either way, the adverbial is commenting on the truth or otherwise of the proposition that the sentence asserts. It is neither certainly nor barely, but arguably. So the adverbial is a modal content adverbial.
Or does its position preceding the adjective mean that it modifies “correctness”? I am not sure. In fact, I am not sure it matters. Either is arguable.