I’m not convinced—and, speaking generally, no one should be convinced—of anything not supported by hard data.
Wouldn't the phrase "generally speaking" sound more natural in the above sentence?
In one sense, “generally speaking” is more natural: it's much more common (according to ngrams for generally speaking,speaking generally). As such, it has a flatter and more cliched tone. “Speaking generally” is perfectly acceptable English, perfectly natural, and should not be discriminated against.
I don't know of an adverb placement rule that applies here. But you may find some of the following of interest: Position of adverbs, bbc.co.uk; Should an adverb go before or after a verb?, ELU; The Royal Order of Adverbs, ELU; and especially Does adverb come before the verb or after the verb?, usingenglish.com.