This is all a great idea but I want to take this opportunity to say as (I think) one of the board's resident Shakespeareans,
we just don't know. Hamnet was 11 when he died, as you say, and that's one of the only two things about him that we know for certain: the other is that he was likely named after Stratford resident Hamnet (also spelled Hamlett) Sadler, a local baker and likely Shakespeare friend who witnessed Will's, er, will. But what an early modern preteen's mental life is like, and where it might have gone had he lived a bit longer, is basically something beyond any reasonable speculation. Ben Jonson, as you might know, lost a son just as Shakespeare did, and wrote an absolutely gorgeous and utterly heartbreaking
poem about it. We know more, I think, about this Ben Jonson, Jr. – Ben had (at least) two sons named that, if I recall correctly – but we still know literally nothing. Did Hamnet play in the theater? We don't know. Was he inspired by his father? Did he dislike him? Was he ever even in London to see his father's plays or his father's acting? Did he even leave Stratford?
We just don't know.
I can accept to some degree the argument that losing a son sent Shakespeare to a dark place – there's every chance it informed that speech of Constance's about losing her son in
King John – but I think it's worth noting that Shakespeare wrote the
Henry IV plays and
Merry Wives probably less than a year – maybe eight months or so? – after his son's death, which, while certainly dark in more than a few moments (the rejection scene in
2 Henry IV brought me to tears whenever I did it) are not exactly the utterly hopeless affairs that are
King Lear or
Troilus and Cressida, the former of which was written likely over a decade after Hamnet's death. I would also add that Shakespeare was capable of writing tragedy (maybe more tragedy of blood style tragedy, but your mileage may very) before Hamnet's death –
Titus is a tragedy, and, in my opinion, far better than the bloody burlesque it's sold as, and
Richard III's last scenes showcase misery to a degree comparable to the tragic speeches of other great heroes (admittedly, the
Henry VIs may be a counterargument because they're just Look What I Wrote, Kit Marlowe!, even if you believe that Marlowe worked on them – which I think I do).
Personally, I don't know if I think Hamnet Shakespeare ever left Stratford. If he lives, the likeliest outcome, I think, of his life is as a middlingly prosperous country squire annoyed at the members of the Tribe of Ben or whatever who keep coming to New Place to gawk at Shakespeare's son. I believe that Stratford fought for Parliament, so probably Hamnet ends his life at Edgehill, if he lives until then. I suspect it makes Shakespeare's works a bit cheerier, but not much so, and probably not for long, but, again, who knows? The arc of Elizabethan to Jacobean drama was likely inevitable (history plays are great but people liked tragicomedies, for some reason, and bloody court plays were fun because you could imagine it happening to James' favorites) but Caroline drama was really wildly working in the shadow of the Bard. That might be where things really do start to change.
The early modern English drama death that would really fucking matter is that of Marlowe's, by the way – if he stays around, Lord only knows what the theater world looks like.
