Fox News owns the extremist images featured at Capitol Hill rally it promoted

The context of this artiicle only supports one definition of the term "own," as used in the title. That is, the colloquial, i.e., culpability. "Fox owns the imagery" cannot be interpreted any other way than Fox is culpable for the imagery. Please, do try and restate that phrase, "Fox owns the imagery" in any other way. That is, any other way that doesn't add all the various caveats you issued in your scenario, but which are not contained in the text of the article. I argued using only what was in the article. Please do the same.
Now, unless you can really restate that damning phrase contained within the title of this article without going outside the article itself, then you must concede that the term "own" refers to Fox's culpability. You can't make the argument, nor do I think you intend to, that the term implicates Fox in the production of those signs, either through physical or intellectual property rights. You can't really say that Fox generated all of the rhetoric contained in those signs, because it's pretty much stock stuff in the conservative blogosphere, hate-talk and right-wing print media. So the term "own" must refer to Fox's culpability viz. the actions of the persons who did produce those signs; that's guilt by association, unless you're really going to try and stick with the custodial obligation metaphor you floated up-thread. In which case, you're just flat wrong. Those people who carried the signs are not dependent upon Fox for the formulation or execution of their actions; they are not drunks in a bar, they are not arsonists seeking tinder, they are not metaphors, they are people. Deeply misguided people, but still people, fully culpable, ethically and legally, for their own actions.
Now, of course, people who perpetrate evil acts concommitantly are equally culpable, but even this meaning disregards the singular concept of "own"ership. Parties can only own equal shares of culpability in equal partnership. The article, by completely neglecting to mention the individuals at the rally, put the "own"ership of the signs in the singular hands of Fox. Again, I ask you, did you not find it conspicuous that the protestors, the actual individuals who produced, brought, and carried those signs, were never mentioned in the article? That the culpability for the messages in those signs was put squarely upon Fox in the very title of this article? You don't find that odd?
So, no, I don't have to address your scenario, because your scenario draws nothing from the article itself. It makes a hopeful inference based upon what you think the author of the article may have meant. It doesn't address the article, but what surrounds the article, namely your disdain for Fox and your urge to defend MMfA. But the article fails to hold anyone but Fox accountable for those signs, signs that Fox neither produced nor endorsed. They are tendered "own"ership of those images through guilt by association. Again, you're very concerned with the matter of truth or falsity, while formal logic simply does not concern itself with such things. It may be true that Fox is culpable for those signs; it is not a valid argument, as presented.