Did the alleged "core feed" exist in the days when there was a breakfiller played out for those regions which didn't opt out for adverts?
Yes - and at one point I think the adverts were inserted outside of the BBC - so a core feed did exist. When the BBC started playing out multiple regional variants I think this changed.
If it was simple to produce an ad-free version - and the incident involving a sponsored bumper going out on BBC One when Breakfast took World News during a fire alarm in Salford suggests it is a tad more complex - then I see no reason not to make it available on the BBC News app for connected TVs. The appropriateness of expecting UK license payers to fund a service we can't see is questionable, and there is some form of precedent in World Service Radio being availabe on digital platforms in the UK.
Where is the licence fee funding? BBC World News is a commercially funded channel that doesn't cost the licence fee payer anything AIUI (unless the channel is making a heavy loss) In many ways this is the same as BBC Brit, BBC Knowledge, BBC Entertainment etc. They aren't provided in the UK either (though are run by BBC Studios - was Worldwide)...
BBC World Service Radio has always been available (and is commercial free) via analogue (648kHz covered a lot of the UK) and IS now largely licence fee funded (previously it was grant in aid funded by the Foreign Office).