You seem to be arguing two different things: first, Fox News does not influence public opinion; second, that it shouldn't matter if they do because adults should think for themselves.

The second is fundamentally a subject view of the world, so you are neither right nor wrong. I disagree with you strongly. We should be concerned about things that harm the our civic health even if we ourselves to blame for them. I think there's a clear parallel to our actual health: when 80% of our population is overweight or obese, it's maybe time to think beyond "adults are responsible for what they eat" and towards "the world would be better if we could make progress on this issue". Ditto our past history with smoking. We know that people are subject to persuasion and psychological manipulation and exploitation of their cognitive biases. A model of the world that has everyone as solely responsible for their own destiny is a pretty useless one, in my opinion.

But, your first claim -- "Fox News doesn't 'swing public opinion'" is an empirical one. Either it does or it doesn't. You're either right or wrong.

DellaVigna and Kaplan's 2007 piece in the Quarterly Journal of Economics investigates this question. They exploit the fact that the phased roll-out of Fox News between 1996 and 2000 impacted 20% of the country. This allows for a natural experiment where some markets are "treated" by Fox News and others are not.

Okay, so, first, should we believe this design? If the entry of Fox News into a market was random, then this is an experiment and we have a treatment effect. If the entry of Fox news into a market was conditional on market characteristics that we can assess, we have "selection on observables", and we can recover a treatment effect. We need only a good selection model. It is also the case that if entry into a market was non-random, but conditionally ignorable with respect to the potential outcomes of the market (e.g. random with respect to politics), we can get a treatment effect without knowing the full selection model. So we have some different routes to the answer here.

The answer is between #2 and #3 -- Fox News did enter markets in a non-random way, but not based on demographics or past voting history, based mainly on geographic considerations. Given the regulatory and infrastructural component of entering a new market, this is probably not surprising, but it does guard against "Fox entered conservative markets first" as a counterclaim.

Now, having clarified the design based considerations, the authors find that Fox News was responsible for a ~0.5 percent vote share increase in Republican vote share in the presidential election of 2000. Whether you consider that large or small depends on your frame of reference. My sense would be that 0.5 percent is small cosmically, but it's maybe possible that you could have an election where that kind of margin is decisive. Remind me again, was the 2000 election close?

This is a fairly credible and careful design published in a good journal and widely cited (1486 citations, which is quite high for a nonmethods economics paper). That doesn't make it true, but it does suggest that the field of Economics views this as an important paper and that it has been exposed to scrutiny from a wide variety of sources.

If there's a little contempt in my reply here, it's because I think your off-hand claim that media diet, priming, and framing effects are solely demand driven and have no actual impact on viewers seems to fly in the face of like 50 years of communications, economics, and political science research. In other words, it feels like your first reply was "50 years of research doesn't matter, humans aren't influenced by any of this stuff". Maybe that's not what you meant, but when I see someone flippantly respond without evidence, I think "Welcome to hell" -- imo, that's the kind of thing we get when we don't teach people to argue.