I finally have mutt configured in such a way that it first tries to display the plain text part of a mail and only the HTML part if there is no plain text available. For years I had mutt configured to display HTML mails using lynx but it was displaying the HTML part even if there was plain text available.
To display HTML mails I was using
auto_view text/html in my .muttrc like it is described everywhere with the following corresponding entry in my .mailcap:
text/html; lynx -dump %s; copiousoutput; nametemplate=%s.html
The problem with this setup is that it displays the HTML part of a mail even if there is a plain text part available. So I had
auto_view text/html disabled for most of the time and edited the configuration file manually to enable it again for the rare cases in which I received a HTML only mail.
But as this is mutt and almost everything can be configured I finally searched and found a solution:
auto_view text/html alternative_order text/plain text/html
If the message has a plain text part and a HTML part mutt shows me the plain text part, but if there is only a HTML part available I get the HTML converted to plain text. Exactly what I always wanted.