Civil liability is an entirely different matter altogether but what's problematic about false advertising isn't that the claim was false, but more that you're breaking a promise. This isn't about free speech, but more about an implicit contract being violated. Political speech, on the other hand, is explicitly what the freedom of speech as a right is about.

Either way, I don't believe anyone would claim that in case of false advertisements, the responsibility falls on the publisher of ads, not the advertiser. It's not generally possible to evaluate the claims made in advertisements - if I were to say, something I'm selling will work for 10 years and will offer replacements if it breaks in the next 10 years, how would the ad publisher know if I would honor the claim several years later?