i mean, kind of a lot, but it's not like this is a wild stance to take. people love to talk about Hamnet's death and how this may correlate to Shakespeare's creative trajectory, particularly, as you might expect, in Hamlet. the issue is that, fundamentally, we just don't know how Shakespeare felt about his children, or the loss of his son, or even how or why his son died. it's a tantalizing blank space into which critics can project a variety of concerns.
comparison: you can also look at the fact that Hamnet was a twin (with his surviving daughter, Judith) and then note how often twins getting separated and then wonderfully reunited shows up as a plot device in a lot of his plays, and then you might think there's Something Going On there. and maybe there was! but also: the divided/reunited twins plot device is something Shakespeare is pulling from classical Greek and Roman comedy, and it was something he indeed used before Hamnet died as well as after. if we're going to imagine Shakespeare as sentimental we can just as easily imagine him as someone who didn't care much about his kids and just liked this plot device for some reason, because at the end of the day, his opinions on any of this stuff were never signaled to the public in a way that survives to us. we just prefer to think of him as nice rather than a jerk, which is fine, but it's still a Choice to fill in the blank.
in the case of the supernatural elements in Shakespeare's plays and whether they correlate in any way to Hamnet's death, we have a similar set of concerns: foremost, that when Hamnet died, Shakespeare had not written very many plays! it was actually early in his career, and even then, a few of things he had written contained ghosts and a bit of witchcraft. Midsummer Night's Dream is definitely his most 'supernatural' play of this early period, and it does date to about the time Hamnet died, but who's to say when he started writing it, and overall does it feel much like a 'dead son' play?
and, moving forward, there are indeed more plays with supernatural elements--but there was also, in 1603, a new monarch on the throne, James, who was deeply interested in the supernatural and witchcraft (Macbeth was essentially written for him directly) and thus that sort of thing spiked in the general culture. the late plays before Shakespeare's retirement, retroactively called "the romances," are certainly very supernatural and sentimental in a way earlier work isn't--but isn't that as much explained by a man getting ready to leave his job looking back on his work and reflecting on stagecraft through the vocabulary of magic? here i'm letting biography determine the reading, too, but just in a different way.
