Shakespeare's popularity with the wider audience rests upon a handful of dramas. Romeo and Juliet is famous for the unthinkingness of the lovers, but the peculiar motivations of the creepy nurse are rarely remarked. In Macbeth, the action is supernatural. Iago in Othello is notorious for his lack of motive (and the same is true of Marlena,) and Othello is completely unreflective. Hamlet is also prompted by the supernatural. Notably Hamlet is such a cipher that one cannot determine his age. Lear's division of his kingdom again is notoriously unmotivated by any intelligible motive.
The construction of the plays is fluid. The dialogue is superb. But what seems to appeal is that the heroes are driven, internally by all consuming passion or externally by malignant magic. They act out restrained passions of the audience. Thus it does not matter that the action often does not make any sense, or that all the theatrics, despite the gloriously thumping words, have no relation to real life of any sort. They evoke a dream life with a pungency usually found in reality. I suppose people will imagine this to be provocation but how is Hamlet, for instance, supposed to be insightful into the human condition when the human condition is remarkably short on ghosts? When you boil the play down to what actually happens, it is astonishing how often the results are manifestly nonsense.
The only comedy widely popular seems to be The Taming of the Shrew. Henry IV survives because of Falstaff. Although to a degree The Merchant of Venice, The Tempest, Much Ado About Nothing, Twelfth Night and As You Like It can occasionally gain an audience, particularly when they or the tragedies are reinvented.
The extent to which Shakespeare's plays served the views of the powerful seems to be negelected. Such things as the witches in Macbeth because James believed in witched, or the foolish Cleopatra because James disliked the memory of his mother are just a small part of it. Henry V is plainly a chauvinist orgy, while Richard III is a Tudor bashing of a predecessor (but covertly of the younger Cecil?) All's Well That Ends Well ends up endorsing marriages arranged for young noblemen (implicitly, by the Queen.) Troilus and Cressida argues against war (feared by Elizabeth lest it give a general enough army to overthrow her.) Alcibiades in Timon of Essex is parallel to Essex.
As to the question of originality and greatness, I think Shakespeare the comedian is vastly more original than is given credit. But without the slighted contributions of Christopher Marlowe and Thomas Kyd, Shakespeare the tragedian wouldn't have existed at all. (Everyone has heard about Marlowe's influence, but Kyd apparently wrote the original Hamlet play, and his interest in Greek tragedy seems to have influenced the Chorus in Henry V.) Those fellows, unlike Shakespeare, fell afoul of the Tudor police state. Shakespeare, despite all the jawing about his humanity and depth, was never in danger of offending the powerful.
Not even when Essex ordered a revival of Richard II as agitation for his planned coup. My guess is that Shakespeare and company escaped unscathed because they informed the authorities as quickly as possible. This perfectly logical explanation is inconceivable to the English patriots who revere Shakespeare, though.
On the notorious question of who wrote Shakespeare, my opinion is that the man from Stratford wrote the plays. The man from Stratford can easily be Shakespeare because Shakespeare isn't such a wonderful person. The guy who wrote Taming of the Shrew could easily be the guy who left his wife the second best bed. As for talent, put up Shakespeare's first six plays against Marlowe's first (and only) six plays.
As to Shakespeare the man, there are some really interesting questions. First, how did he get the money to buy into a partnership? He wasn't a famous actor and no other Elizabethan or Jacobean playwright managed to make much money.
How would he know the official version of Marlowe's death (alluded to in As You Like It,)? The death of Kyd, Marlowe's roommate, was apparently from lingering effects of torture aimed to elicit charges against Marlowe, was enough to cast doubt on the official story. Why, then, did he trouble to propagate it to an audience not interested in the private lives of playwrights? Probably Shakespeare kept track of Kyd, apparently basing his Hamlet on Kyd's earlier play. And where did Shakespeare find a copy of Kyd's Hamlet, inasmuch as the authorities had confiscated Kyd's papers? Playbooks were proprietary, and even Shakespeare has problems with plays being lost.
The obvious solution, that Kyd's Hamlet was not a significant influence, doesn't seem to work, given Hamlet's odd structure. (Theatrical producers tend to cut Hamlet savagely. Branagh's version was notable most for actually playing the whole damn thing.)
Where was Shakespeare's library when he died?
Did he die of syphilis? (Read the last two sonnets.)
Incidentally, I see no reason to think he was gay, or even bisexual. One sonnet very clearly denies this. Shakespeare the flatterer seems to me to be so obsequious that he would gladly be some extreme in his devotions, especially to a hoped for patron he thought might be effeminate, in his poetic tastes at least.