The point is that the ads are a waste, and that the advertisers aren't getting their money's worth. It's really remarkable that they still haven't caught on to this.

The ads are not a waste. It's called remarketing and it does boost conversion numbers over raw untargetted ads. The ads are targeting people who went to a site, browsed around, didn't buy but might do in future (much more likely than a random unselected person).

Generally the ad network does not know if you really completed a purchase or dropped off during the funnel. Presumably advertisers could set up a hook to inform the ad network if you did purchase or not, but I guess they don't care enough to do so.

Is there even a way to check if the google and facebook charge the correct number of advert impressions? Seems to me like they can print money by reporting higher numbers.

I haven't played that game in a long time, but back then they were the sole determiners of whether a "click" was valid or not. I got charged for clicks for which people only stayed for a second or so. Now there were a lot of people who claimed they can make a decision about purchasing from a website in a second, but I don't think they're thinking it through. I think they were erroneous clicks or competitors, yet I was charged.

They may have improved.