> site showing the ads could easily fake
Yes, that's what I mean by "unable to verify"; If the site is honest, everyone would be just as happy, but there is already so much fraud, even with (relatively) trustworthy Google and Facebook, that everyone wants some ability to verify; If all the traffic goes only through the site, that ability is lost.
> Assuming that the advertising site would just have a proxy that tunnelled everything to the ad server, would the ad server not get the exact same traffic?
Not exactly - the wide-sense-cookies[0] that allow the ad server to track users across different web sites would not be. Fingerprinting[1] would still work subject to advertisers trusting the proxy.
[0] including flash objects, etags, localstorage, etc, see https://samy.pl/evercookie/ for a 7-year-old discussion.
[1] https://panopticlick.eff.org/