Most of everybody thinks their behaviors and decisions are not meaningfully influenced by advertising. Companies spend literally trillions of dollars running ads. One side is right, one side is wrong.
And advertising largely relies on this ignorance of its effects, or otherwise most of everybody would go to much greater lengths to limit their exposures to such, and governments would be more inclined to regulate the ad industry as a goal in and of itself.
Advertising is just companies saying "This is what you can purchase from me - it's awesome - please consider purchasing it". I have managed hundreds of millions in ad spend for major brands. None of them rely on weird ad magic to persuade people secretly - just showing off different aspects of the product or service.
No, this take is crazy. If ads were able to brainwash people Coca-Cola would still be the most popular drink in America.
The problem with "meaningfully influenced" is that a 1% bump in sales is massive for a company, but normally only represents a very minor shift in customer behavior.
US spending on advertising is, in total, about $1200/person-year. If you believe that advertisers are rapacious capitalists who will take as much as they possibly can, then they only believe that they can capture about that much extra per person by advertising to them.
That's not nothing, but it's not very much either. Ads are extremely overblown as a threat to society; you only need to look as far as eye-tracking studies of web browsers and the prevalence of ad blockers to see pretty good proof that people do just ignore them most of the time.