The purpose of "free speech" is to allow the spread of ideas.
The purpose of advertising is to spread an idea.
They are different things.
Yes, the purpose of "free speech" is to allow the spread of ideas. The purpose of any particular piece of speech (a book, a pamphlet, a poster, a sign, a rally, a concert, anything) is to spread an idea. The idea in that particular piece of speech.
Do you want to preserve free speech but ban speech that tries to spread an idea? Your comment would be banned because you're trying to spread that idea.
Commercial speech is a legal term for speech that promotes commerce [1].
Trying to ban all advertising of course wouldn’t get anywhere (especially under the current SCOTUS), and the article’s author clearly hasn’t really thought through the implications. But there is a legal door that is ajar, in the Central Hudson test, and could potentially be widened by arguing that some classes of today’s advertising are against the public interest; the First Amendment is already not blanket covering all commercial speech.
Meanwhile, even the original Central Hudson test --- if commercial speech concerns lawful activity and isn't misleading, the government (1) needs a substantial interest and (2) must narrowly tailor the restriction --- was inhospitable to the sentiment in this blog post. But Central Hudson has as I understand it gotten more restrictive, not less; for instance, Sorrell, which was a bipartisan decision, applies heightened scrutiny to regulations under Central Hudson.
You can keep cigarette ads out of Highlights for Kids. But you're not going to be able to keep Nike from buying up ad inventory, and you'll certainly not be able to regulate political and cause advertising (and "propaganda").