tl;dr I believe this is about encouraging accuracy, proportionality and careful attention to materiality in reporting, not stifling debate and discussion.
This is really heading into "The Park" territory.
I don't think anyone disagrees that a media outlet has the right to set its own editorial policy, including political standpoint. What is, in my view, unacceptable is when the media report stories in a way that goes beyond what the evidence supports, often including the misuse of statistics. This is often done in ways that the proprietor and editor knows will resonate well with the readers, who might not read their familiar newspaper that critically. Unfortunately, balanced reporting of the issues and acceptance that we live in a complex world does not make for "juicy" headlines that sell papers.
There is an unfortunate tendency for every story to become polarised in the way it is reported. Take an issue that has personal resonance with me because I am severely physically disabled - the Paulley v First Group
case about wheelchairs and baby buggies on buses. The case itself was about the right way to balance the needs of different users when there is only so much space on the buses, especially how this balance reconciles with the company's duty to provide reasonable adjustments under the Equality Act 2010 and the limits on the driver's powers under the regulations governing Passenger Carrying Vehicles. Much of the reporting seemed to want to turn this into wheelchairs v buggies, almost like a Harry Hill style punch-up. It wasn't about who "wins", but about how to try to ensure everyone can travel. There is no easy answer; if there was, the case would not have finished up in the Supreme Court.
It is arguable that the best answer is to introduce buses with separate provision for wheelchair users and buggies; such buses already exist in some areas. Somewhat analogous is the best practice of separating accessible toilets and baby change facilities. Partly this is because users can need a fair amount of time, partly this is because some disabled people have limited ability to wait and partly this is for hygiene reasons. Nappy changes and the use of potties can be incompatible with self-propelling wheelchairs (where anything on the floor will likely land up on the push rims and therefore the person's hands), also those reliant on clean techniques such as intermittent catheterisation are particularly vulnerable to any faecal contamination. However, it is still common to find the baby change in the accessible toilet, even in high traffic areas like motorway services. If I come across such a setup, I often find people are embarrassed if they feel they have made me wait in my powerchair or that they do not have the right to use the facility. That's not the way it works - there is no absolute priority for any group of users, nor do you have to disclose your (potentially hidden and embarrassing) reasons for wanting to use the accessible facility. If you have a need to use the facility, use it. I'll wait my turn if I can, otherwise I will ask politely if I can go next.
There are relatively few ways to call media proprietors to account over accuracy. The media, with considerable justification, has successfully argued for light touch regulation because of the importance of freedom of speech. A healthy democracy relies heavily on a free media to call government to account. There is a recognition that the right of free expression under Article 10 of the European Convention on Human Rights should not readily be curtailed. Yet, the power of the media has been abused, as was seen clearly during the Leveson inquiry.
Media outlets rely on funding - and in an increasingly online world, a lot of that funding comes from advertisers. Nothing in this campaign is suggesting the curtailment of discussion and debate, merely that inflammatory reporting of sensitive issues accompanied by commentary expressed in highly emotive terms is unhelpful and might be something advertisers wish to dissociate themselves from. In today's society, money talks - and it is an entirely legitimate form of activism to lobby economic actors to use their power to bring about specific ends, whether that is pension funds disinvesting from environmentally damaging companies or advertisers deciding not to advertise with certain media outlets. The effects of mass media editorial policy affect the whole of society, not just those who consume a particular outlet's output, not least through the ways that the media seeks to influence people's political views.
As has been said by various commentators, ask a representative group of British Muslims what they feel about the Daily Mail's reporting and the effect polarised reporting has had on their lives. There is no direct cause and effect between a particular media outlet and hate crime, but that does not mean unjustifiable polarisation in reporting should be socially acceptable. That said, we should not resile from reporting abuse by minorities honestly - but we should be careful in the way we use labels. If I was found guilty of a serious crime, then that should rightly be covered by the media and that coverage may well include pictures. However, it is immaterial that I'm disabled (or, for that matter, white, male and Christian), so such matters should not really feature in the commentary.
Materiality should always be a consideration for journalists. We should be particularly careful about confusing religious practices, cultural practices and individual beliefs. In situations like the Rotherham sexual abuse cases, there is understandable outrage over what the guilty did. Yet, if you are going to identify some or all of the guilty as Muslims, surely you should give equal prominence to the total condemnation of what they did by the Muslim community in Rotherham and beyond? My personal feeling is that the religious affiliation of the guilty is immaterial; I can't think of any mainstream religion that regards sexual abuse of the vulnerable as appropriate. Even for those who try to read news reports critically, it is hard to resist unconscious prejudice if a link is suggested to you.