This isn't a simple binary problem; there are levels to consider along the lines of this (overly simplistic) distinction:
- Words that are merely meant to be insulting, disparaging, bullying, degrading, etc.
- Words that are politically problematic: that carry the weight of social oppression, historical group violence, or deep-seated cultural hatred.
If we use terms of the first sort — like 'fatty', 'nerd', 'd_kwad', 'f_khead', 'sh_t-for-brains', etc, — we hurt people's feelings — sometimes extremely badly — but at the end of the day it is personal and individual, without any real social or political teeth to it. People get called names and they (usually, not always) survive and grow. It's a nasty moment in human life, sometimes even a fatal one, but it's a moment that can pass.
However, when we use terms of the second sort — as people in the US sometimes do for blacks, Jews, Muslims, women, gays, and others — we're not just hurting feelings. We are echoing the language that mobs of white men used during lynchings and slave auctions; that Nazis used while rounding up people for concentration camps; that men use during sexual assaults; that Right-wing pundits use to stir up anti-Islamic hatred. It is not the language of a person degrading another person; it is the language of a society degrading a group, and that group degradation continues as long as the language is kept alive.
Respectable news outlets and other forms of mass media do not spell out or use the n-word (or similar terms), not because they're worried they might offend some particular person, but because the use of the word carries the weight of social oppression. If news outlets regularly used that word, we would be right back in the mindset of the segregationist South, where whites are 'whites' and blacks are 'n_s', and any possibility of equality or fair treatment fades away like the morning dew. Instead, respectable news outlets prefer to talk about 'people', without unnecessary reference to stigmatizing language.