Did the word didactic go through Latin before arriving in English?
How could it not have?
Some people learn Greek. ("I would make them all learn English: and then I would let the clever ones learn Latin as an honour, and Greek as a treat." — Winston Churchill).
Yet Websters says it came to English directly from Greek!
It is correct.
I think they are wrong.
Why? Can you cite it being used in English prior to Milton, and then demonstrate that this source was familiar with Latin, but not with Greek?
There is a Latin word, didacticus, which means didactic and was widely used in Latin texts.
And Danish has didaktiske, but that doesn't mean it came into English from Danish.
How can we find out what is didactic's true etymology?
It came into English as a coinage of John Milton's (and a true coinage, as διδακτικός means "apt at teaching" rather than "a didactic author or treatise", as he used it—the adjective sense coming later).
Milton knew Greek, he wrote in it and about it (in particular his idea Christian tragedy should combine the legacies of Greek Tragedy and Hebrew Scripture).
With his strong command of both Latin and Greek, in turning to them for a word, he would have known διδακτικός and needed no intermediary.