It's not a strawman if it's true.
I don't think anyone is suggesting Fox News be shut down (probably unconstitutional anyway, and since cable bandwidth is unlimited, it is immune from a Dem fairness doctrine attack anyway). I actually think there is some value in having an assortment of hyper partisan media. So why don't we just stipulate that Fox News is an RNC partisan shill operation and move on?
The problem is that under the current system ('private' ownership of media) only right-wing viewpoints can be heard. If the media is entirely owned by the ruling class we will never hear any representation of the opinions or interests of the masses (except of course as those opinions benefit the rulers).
In other words the problem is the entire system, in which 'freedom of speech' is impossible, not just FOX news.