semi-convoluted analogy time!
I want to start off with a statement of empathy and understanding for people who have their name slandered online when someone reviews their shop/eatery. Now lets say someone comes onto that shops site, and leaves a bitter unsightly comment, I don't think anyone would argue that they are within their right to delete that from their site, as much as they could leave it up and respond to it at their discretion.
Now lets say, that someone comes onto your site, and leaves a comment about this persons shop. your site could be a review site, or just a general discussion site. Either way, when the shop owner comes to you and asks you to remove the comment, you have every right to do so, or to decide not to.
If I see someone come along and FORCE someone (you) to take down a review or comment, I would be very upset about that. I mean, its not my site, but when I see your rights get trampled on, even though I empathize with the trampler, I have to take your side on this.
All this to say, people who take obvious black and white issues and use those to further their agenda do not reserve the right to be unchallenged in process.
We all agree that killing people is terrible, but we don't all agree that abortion is wrong. We all agree that giving to people is great, but we don't all buy into a universal basic income. We all agree copyright protection is there for the artist, and that stealing things is bad, but some people think that the argument of what constitutes data you can control and can't has farther reaching consequences than piracy.