Otherwise you're just asking for trouble when the same user who signed up with an email address at maré-design.fr later tries to reset their password with an email address at xn--mar-design-d7a.fr. Sorry, you don't seem to have an account with us.
The same thing happens with URLs. Some browsers send the Host: header in punycode but send the referer in UTF-8. Who knows how they encode CORS headers and all the other newfangled stuff that contains bits and pieces of URLs. You have to consistently convert one to the other before using any of them.
IDNs are a mess.
I don't think <input type="email"> does that though. Maybe it should, but users might be surprised to see their address automatically turn into some ugly unreadable xn--whatver-d7a domain. After form submission then yes of course any sane process will convert them to a canonical form.
I know there were good reasons to use punycode instead of UTF8 for IDNs, but it sure is a mess.
Ditto for internationalized TLDs like something@taiwan.台灣).
I'm not sure if this is really that bad, but I don't like it.