This isn't a "press" problem, its a deeply ingrained social and cultural one.
Any illness, affliction or other deviance from the healthy normative is either accepted by society, or it isn't. Name it. Leprosy, AIDS/HIV. It takes a social perception shift for these things to be accepted with a sympathetic ear, until then sufferers are ostracised by the society they live in. If, god forbid, Gazza was struck with a form of cancer, opinion towards him would change in a heartbeat.
The fact is that mental illness and addictions are still stigmatised socially. It's not well understood by the community at large, and therefore is seen as an abomination. People (in the collective sense) will justify things like alcoholism as 'not a real illness', the person is 'just weak', or blah blah blah. Ultimately, its a form of self protection, revelling in it in a form of "I'm better off than you" self-valuing mindset, holding up public examples of the likes of Gazza as a benchmark to validate why things could be worse for themselves personally. The faux-pity isn't a genuine concern for the sufferers well-being, its a water mark which says "awe, you're in trouble, thank goodness I'm not down there at your level".
Hence why its so easy for the press to pray on someone like Paul Gascogine; it feeds a social stigma. When he eventually passes on, he will be celebrated and revered. Until then, society will mock and devalue him.