How can I get the DOM to reflect the modified input value?
I'm sorry, but the problem is not in the example code you supplied. Or when this is the only html in your page, then you're missing the starting and closing tags.
This clean page runs perfectly on my machine:
Or are you testing in Internet Explorer? In that case, remove the line which is logging to your error console:
console.log('div.html(): ', $('div').html())