The argument is that while you can’t program with it everyone has access to it therefore it’s a programming language.

That seems weak.

Pen on paper has the same result as HTML and is arguably more accessible. Does that make keeping a diary “programming?”

Ultimately it doesn’t matter, but there are significantly better arguments than what was presented here.

The question really should be “is a markup language a programming language?”

The people who argue about this probably bikeshed about a lot of other insignificant details.