One of the troubleshooting steps mentioned in the wiki article I referenced suggests that you enable email archiving, and install the email archive manager addon.
That'll show you that the emails are in fact getting sent, and what format they're in, including their content.
Basically, this will simply prove that the emails were "sent". ie: they were handed off from Zen Cart to your mailserver without any error being reported to Zen Cart.
What the mailserver does with the message after that, including stripping off HTML portions etc, is up to the mailserver. Usually this issue needs to be sorted out by the server administrators.
One reason for testing alternate Transport methods is simply to try a different way of delivering the message to the mailserver. Sometimes the mailservers are set up to strip off sections of messages that they think might be at risk of containing spam, esp when they come from the webserver. Sometimes this can cause the entire message to get dumped, or just the HTML portion removed.
Sometimes telling Zen Cart to "send from known domain" will help since it sets the "from" address to the same as your store address ... which ideally would be an email account in the same domain as your store itself ... thus making the email look very legitimate and not flagged as potential spam.
That's why those ideas are mentioned in the email troubleshooting guide I referenced.
If none of those ideas are working, and you've checked that your server isn't blacklisted in some way, etc, then you really need to take up the issue with your hosting company and ask them why your messages are being mangled before they get delivered.
