First off, Chris expand your acronyms. LaTeX is sufficiently well known there's no need for that. The Text Encoding Initiative, (which doesn't seem to be an actual encoding standard?), despite being active since the 1980s, and my own use of various markup systems since about the same time, was unknown to me:

https://en.wikipedia.org/wiki/Text_Encoding_Initiative

Even knowing that, it's not clear what problem exactly TEI solves.

Second: what's your goal? How are you looking to use Markdown?

For offline use in your own document preparation activities, I think Chris answered his own question. Markdown is extensible via HTML. If you want to use semantic elements such as <em>emphasized text</em> or <cite>Some Famous Book</cite>, then simply include those elements within your own source document.

A lot of us are using Markdown on third-party sites -- for myself it turns up on Reddit, Ello, Diaspora, and StackExchange ... in a mazy of twisty passages, all different. And I really don't have control over what's parsed or supported (Ello's lack of blockquote and Reddit's lack of images both drive me equally nuts).

To a large extent, for works of a few dozen to 10,000 words, possibly even longer, heading structured documents work great. Markdown is just that: a quick and only slightly dirty way to create content for which presentation specification (e.g., bold and italic specs) is more than sufficient. I mean, yeah, it's fucking wonderful that HTML has, in theory, all this wonderful semantic markup. Dig into the source of pretty much any site out there, with very few exceptions, and you'll find they're simply butchered. Table layouts and <font> directives for titles and headings and absolute positioning all over the page.... Or worse, custom styles which make even <i> and <b> look stunningly attractive.

If anything, documents tend to go to the other extreme: too many entities, too much markup, too much crud, too much design. To the point that I'm far too frequently scraping raw text from documents and rebuilding them (usually with Markdown) to a simple basic PDF or HTML page that I can actually fucking read.

OCD, it's a hell of a drug.

My suggestion: get over it.

You cannot get both a SIMPLE and a COMPLETE markup language. One is the enemy of the other.

Settle for good enough.

The problems I've got with Markdown (or any of the other lightweight markup languages: https://en.wikipedia.org/wiki/Lightweight_markup_language) is that standards aren't, parsers and interpreters (particularly on third-party sites) vary and/or suck, and stuff I'd really like to see (superscript, subscript, footnot, ToC generators, equation support) aren't well supported. Hell, we can't even rely on blockquotes, images, character escaping, and superscripts being uniformly treated.

Use Mardown. Or switch to another LWML. And extend it with HTML. Use post-processing scripts to add those div wrappers and metadata you want (I'm basically doing the same for my own work).

The one other element that might make sense, and which you don't address, is a document standard -- I'd really appreciate if documents had standard metadata: title, author, publication date. Something vaguely like an RFC #822 of text document metadata (and fortunately we've got *so many standards to choose from: https://en.wikipedia.org/wiki/Metadata_standards -- Ob xkcd 927).

It's also not clear how or why LaTeX itself isn't suitable here. It's plaintext, more structured than Markdown, yet generally easier to code than HTML.

I do appreciate the tip on ScholarlyMarkdown, which looks like it might address a few of my own interests.