C# Send both HTML and Text email - most elegant?
I would say that, in today's world, the "best-practice" approach would be to ensure that you send your message as both plain text and HTML (if you really want to send HTML email messages).
Oh, and make sure you do actually send the content in the plain text view, rather than a single sentence saying "You must use an email client that supports HTML messages". Google Mail takes this approach, and it seems to work perfectly, allowing "rich" views on full-fledged PC clients, whilst also allowing "minimal" views on more restricted devices (i.e. Mobile/Cell phones).
If you want to take a purist's view, you wouldn't be sending HTML emails at all, nor would you ever "attach" a binary file to an email. Both corruptions of the original email standard, which was only ever originally intended for plain text.(See some people's opinions of this here and here)
However, in the pragmatic modern-day real world, HTML email is very real, and very acceptable. The primary downside to sending HTML email is whether the recipient will see the email in the way that you intended them to see it. This is much the same problem that web designers have battled with for years; getting their websites to look "just right" in all possible browsers (although it's significantly easier today than it was many years ago).
Similar to ensuring that a website functions correctly without requiring Javascript, by sending your emails as both HTML and Plain Text, you'll ensure that your emails degrade gracefully so that people reading their emails on (for example) small mobile devices (something that's becoming more and more prevalent these days - and which may or may not be capable of rendering a complete HTML email) can still read your email content without issue.
If you only send HTML, then anyone reading email on a text-only device will have trouble.
For example, I suspect many low-end mobile devices are capable of reading email but not displaying full HTML.
I would say it's best practice to either send text-only, or text and HTML.
I don't see why you're catching a bunch of exceptions only to rethrow the same exception type with a different message, by the way - the original message may well be more descriptive.