Current html output of realfavicongenerator is as follows:
<link rel="shortcut icon" href=".../favicon.ico">
<link rel="apple-touch-icon" sizes="57x57" href=".../apple-touch-icon-57x57.png" />
<link rel="apple-touch-icon" sizes="114x114" href=".../apple-touch-icon-114x114.png" />
The first tag is not self-closing, but others are.
There is better not to use self-closing tags at all, browsers are parsing response as xhtml, only if it's Content-Type be application/xhtml+xml (which is very rare). If the content type be text/html, the document shoud be HTML not XHTML.
You're right, this is a bit messy and not accurate. Thank you for reporting!
The update is easy, testing will require more time. I will address this next week.
Done and deployed. See http://realfavicongenerator.net/change_log