Josh,
To prevent this thread getting out of control and getting to the point where everyone is name calling, let's step back and clarify a few things. While I respect your opinion that paid search could influence traffic from a personalization standpoint, your explanation leaves too many questions unanswered. I personally won't trust anything someone says without a valid explanation WHY they have the position they have. If you can produce a few links/references so we can take a closer look at what you're talking about, that would benefit this discussion way more than simply saying "because I talked with X, and they said it's true." Sharing more information can help educate everyone here, and help us all grow as professionals.
I really don't care who feeds information to whom; the only thing I care about is the fact. The fact is (according to my knowledge) Google has stated paid search (Adwords) does not directly affect organic rankings.
Google Forum thread disputing that PPC influences rank: https://productforums.google.com/forum/#!topic/webmasters/tN5pIG8Qy0I
Google's Search Console Help: https://support.google.com/webmasters/answer/35291?hl=en - "Advertising with Google won't have any effect on your site's presence in our search results. Google never accepts money to include or rank sites in our search results, and it costs nothing to appear in our organic search results."
I agree that promoting your site through a variety of mediums will send a signal of trust that your site is legitimate, but that would be a macro view that everything helps a little bit. The question to this thread is, "Do you really think that them pointing their ads to the actual domain is going to increase your rankings?"
That answer is No. Pointing ads to the actual domain will not directly increase your organic rankings. Will it help with branding and trust, sure. But that's not the question we're trying to answer.
Now... If you have evidence that contradicts my points, I'd be more than happy to discuss and debate. I think opposing viewpoints are important to the development of the community/industry, and is really what makes it exciting. However, if the only thing anyone has to say is "I heard it from so and so" then this thread deserves to be closed if there's no additional facts to support an argument one way or another.