...works of propaganda that are nonetheless brilliant, compelling stories.
Perhaps then we should reconsider our notions of brilliance and what makes a compelling story?
Shakespeare pioneered the use of prose instead of verse. Shakespeare also progressively eliminated the use of clowns in drama. His personal evolution strikes me as being impelled by notions of naturalism or realism that are unavoidably less developed than ours, but are still efforts to be more faithful to his vision of humanity.
Within the proper political limits. There is however not the slightest indication that he had any problem with advocating any and every policy of those in power. Shakespeare's relations with the powerful was so strong that he and the Globe theater survived Essex's rebellion unscathed. Even though Essex had arranged a showing of Richard II (as I recall) to prime the public! Personally, I suspect the company informed to the government as quickly as possible, which is why there was no political discomfort.
I can only suspect this as someone bemused by the insistence that Shakespeare is the inventor of the human. That seems to be the real position of most Shakespeare lovers, even if they are more discreet in their wording than Msgr. Bloom. (A marginal note: One of Harry Turtledove's halfway decent books, Ruled Britannia, was inspired by this incident.)
The memory of highly stylized, non-realistic morality plays was still extant in Shakespeare's. Really, I've wondered if Othello isn't in many respects a hidden morality play about envy, with sexual jealousy thrown in for special thrills. At any rate, the trend of the Elizabethan and Jacobean theater I think was definitely aiming at more naturalism and realism.