I suspect their proprietary widget includes a converter.

All Amazon sees is a suitably acceptable legacy MOBI. No EPUB, AZW3, Docwhatever, et cetera and so forth.
You should be able to configure the send-to-email addresses in calibre with a list of acceptable formats -- just specify MOBI as the only format, and calibre will automatically convert to MOBI if necessary, before sending to Amazon. Just like ereader profiles, which also specify which formats they accept.