The courts already distinguish "commercial speech" as a class of speech. Would we prevent all forms of commercial speech? What about a waiter asking you "would you like to try a rosé with that dish? It pairs very well together." Is that "advertising" that would need to be outlawed?
What about giving out free samples? Is that advertising, and thus should be illegal?
What about putting a sign up on your business that says the business name? Is that advertising?
I hate advertising and propaganda. But the hard part IMO is drawing the line. Where's the line?
You are starting from technicalities before you even took the moment to actually think of goal is worth it.
If it is worth it and we would indeed be better off - plan how to go there, it may not be easy, but it will be possible this way or another. If it is not worth it after all - just say it, no technicalities needed, redirect time and effort elsewhere.