In a way that is effective, helpful, or beneficial; in a manner that successfully avails or accomplishes something.
From 'availing' (present participle of avail) plus the adverbial suffix '-ly.' The layered suffixes create an adverb meaning 'in an availing manner.'
Like 'availably,' 'availingly' is grammatically sound but virtually extinct in modern English—it's an example of how English speakers prefer clarity over morphological correctness, shedding unnecessary adverbial forms.
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