Unlike others you've mentioned, MIME is not a rich text format by itself. While MIME and "Enriched text" are part of the same set of standards designed specifically for email, HTML is a markup format initially developed for the Web by the W3C.
However, nowadays most email clients and services use HTML for rich text formatting. This is because email clients and webmail services can re-use existing HTML rendering engines for displaying email:
- webmail is in your browser; it uses your browser's rendering engine
- Microsoft Outlook (at least some versions) uses Internet Explorer's engine
- Apple Mail uses WebKit, the engine powering Safari and Chrome browsers
- Mozilla Thunderbird uses Gecko, the same as Firefox
HTML email also allows for styling with CSS, enabling professionally designed email campaigns.
Therefore, HTML format is used when you use the "Rich formatting" mode for email. It is well-supported in all popular email clients, including webmail. It is the CSS support that varies wildly between clients, especially in webmail where different services (e.g. Gmail, Hotmail) force different limitations on what can be styled and in what way.
I think the reason why "Enriched text" has not seen wide adoption is not that HTML is "better" for email (there are too many reasons why HTML is not well-suited for email to list here), but because people used what they were already familiar with and what they have existing rendering engines for.
There is something else to add here. Not all email clients run in a graphical environment; there is a significant portion of people who prefer to read email from terminal emulators ("from the command-line", in other words). Therefore it's important that all email also carries a plain-text format of its contents. Regardless of whether you use the "Rich formatting" mode while you compose email, Gmail always includes the message body in plain text format. I suspect other modern email clients do the same, although I haven't tested.
When email carries multiple formats—HTML & plaintext, in our case—it's up to the email client that receives it to choose which one it will present to the user. Graphical email clients will render HTML by default, while text-only clients will ignore the HTML version and just show the plain-text one.