There are a couple of issues here. First, Markdown and HTML simply punt on the vast majority of the issues that TeX solves. Just as the author rightly comments that TeX is not geared toward online publication, HTML is geared toward only that model. If you want to paginate HTML or Markdown, you do it yourself. Widows and orphans are (obviously) your problem to deal with. Compared to the work that TeX is doing, Markdown is, to a first order approximation, just catting the file. HTML can reflow a document in real-time because it's doing a really poor job of reflowing the document. Even when they work, they're just putting line breaks in whenever the width would otherwise be too wide. TeX is running a dynamic programming algorithm to minimize the "badness" of the line breaks across multiple lines and even paragraphs. And quite a lot of the time, the browser just throws its hands up and says, "fuck it, you can just scroll horizontally to read the rest of this line". You can't do that on paper. So of course it's faster. You might as well be complaining that Preview is faster than Photoshop.

HTML and Markdown don't do automatic hyphenation (across multiple languages). They don't do ligatures. They don't do proper text justification (neither does Microsoft Word or Libre Office for that matter). They don't do cross reference tracking (i.e., having automatically numbered sections, tables, figures, etc. with automatically updated references). They have no logic at all for automated float placement. Font handling is specified by a human instead of relying on algorithmic font selection and substitution when necessary. I could go on for pages of this.

I think the idea that web browser vendors are better at this sort of thing than TeX and LaTeX is so wrong I don't know where to start. The author complains that some of his 20 year old LaTeX articles rely on outdated files to render properly. While this is true, and very occasionally a problem, it's only very recently that you had even the vaguest hope that your HTML document would render the same way on two different computers owned by the same person today! Arguably, the biggest slice of the software industry is now devoted to making things render on browsers. And for Markdown, we quite recently saw that even the simplest text rendered in no fewer than 17 different ways depending on which software (and version) you processed it with. If my goal is to be able to reproduce the output of today 15 or 20 years from now, HTML would be the absolute worst choice I could think of, unless again, you stick with <b> tags and the like, and the subset of LaTeX you can reliably assume will always work gives you much broader coverage of the space of typesetting issues than the subset of HTML that doesn't change monthly does. Not to mention, I can still more easily go get an old LaTeX implementation to rebuild a document that doesn't compile anymore (but in 15 years, I've never had to). It's quite a lot harder to get Netscape Navigator 3 up and running to attempt to faithfully render a document I wrote in 1997.

Also, web browsers have historically been just about the buggiest, most insecure, and transient pieces of software we've ever written as a field, and TeX is famously maybe the highest quality piece of software ever written. It's more or less fine that the web changes every 18 months. It's a problem for archivists, but the web isn't really intended for that. Academic publications are though, and the impedance mismatch is, in my opinion, brutal.

The interface (by which I mean the programming language) of TeX and LaTeX is indeed pretty dreadful, but this is a really minor issue compared to the rest of it. There are a lot of things I dislike about LaTeX, but I don't see how HTML or Markdown is an improvement. You'd need a completely new thing that supported everything that LaTeX supports, and while you could certainly do so with a nicer language, you couldn't do it with something as clean and simple as Markdown -- there are just too many things you need to be able to tell it you want it to do.