It's about use of the data, not its storage
> However, someone who has committed no major crime - or merely done something embarrassing - should usually be allowed to have it forgotten at some point rather than having the incident follow them around on the internet forever.
This, I think, is the nub on which nobody has put a definitive finger. The BBC, and other news organisations, including El Reg, collect personally-identifiable information as part of news stories. They process that data, store it, and reproduce it on request, which is just how the WWW is supposed to work. Some of that information will be more or less embarrassing to the data subject, but the news organisation has a continuing need to process it. 'The "Right to be Forgotten" only applies where personal data storage is no longer necessary or is irrelevant for the original purposes of the processing for which the data [were] collected.' 
Where it goes wrong is when another party, say a prospective employer, or an insurance company, retrieves the data and misuses it. Then, that other party is at fault - not the news organisation. At that point the data are being processed *by the other party*, and that must be done in compliance with the Data Protection Act. If the data are irrelevant for the purposes of deciding a work appointment, or issuing an insurance policy, then they cannot be used by the employer or the insurer for that purpose.
I will admit that preventing the misuse of retrieved news is harder than suppressing the news, but I'm in favour of the former, and firmly against the latter.