If we presume that it’s a technical limitation on their end, albeit sad, they could communicate…
Dustin Diaz

Thank you for the quick reply, Dustin. I completely agree with what you wrote (which is fairly close to what I would have written). It implies “even though circumstances are difficult, we are listening, and we care.”

For what it’s worth, here is the response that I got (within a stunning :05 seconds of hitting submit):

We do not offer username changes at this time.
We also do not free up inactive, suspended, or deleted usernames. This includes old justin.tv accounts.
You can use your email account for multiple Twitch usernames, however, if you are interested in closing your existing account you can do so via your Disable Account page.
In the future we hope to offer a service for username changes, but as they are used as critical fields in many of our databases, this will take time to plan and execute.
If you are simply trying to change the capitalization of your username in chat, you can do so via your Profile / Display Name settings
Best wishes, 
Twitch Support