> You can include arbitrary HTML tags in Markdown at any place you need them.

That is well known and I am sure the author is aware of it. The problem they are describing is not whether HTML is technically allowed inside Markdown. It's that when you are writing Markdown, you are writing Markdown, not HTML, and that comes with some problems.

> It is perfectly capable of what the author claims it isn't.

In theory, yes. In practice, using Markdown becomes much less appealing once you start dropping raw HTML all over the place. The whole point of choosing Markdown is that you do not want to spend your time typing <p>, <a>, <li> and the rest. You want to write in Markdown, with only occasional HTML when absolutely necessary.

That is exactly where the author's complaints become relevant. If the solution to Markdown's limitations is routinely switching to HTML, then the argument becomes circular. If you are expected to write HTML to address the author's complaints, why bother with Markdown at all? If the answer is just "write HTML", then you may as well skip Markdown in the first place.

Most markdown engines allow short tags to stand in for html, so for frequent features you can just use a short tag.

Alternatively you can extend markdown. I wrote a simple text based game engine that was markdown based but I needed some arbitrary additions appropriate for a game.. so I just added a few elements.