>If someone clicks your ad and buys the product during that session, you know the ad worked

This is not as clear as you make it seem. Let's say there's a hypothetical product that a consumer only buys once every 5 years. If someone clicks your ad and then immediately buys the product... what if they would have bought it anyway, tomorrow, or 5 minutes from now, without the ad? How can you test that counterfactual?

If I buy cat litter online once a month every month for 5 years by going to example.org/catlitter -- but then they decide to start advertising on facebook, so now I click the facebook ad once a month to buy cat litter from the same site, are the ads "working"?