For the HTML email that doesn't work as intended (which is not a lot, mostly newsletters, which I prefer to read via RSS), I use https://github.com/tshirtman/.mutt/blob/master/mutt_bgrun and this macro. It picks up the first HTML attachment in the mail, saves it to a local file and opens it in a Firefox tab (and Firefox is already typically running):

macro index,pager ,b "<view-attachments>/html<enter><view-mailcap><exit>" "View first HTML attachment in browser"

You'll need to combine it with this entry in .mailcap:

text/html; mutt_bgrun /usr/bin/firefox %s >/dev/null 2>&1; needsterminal