This is a subject that was actually debated in court (the Coughlan judgment) and that's why Continuing Healthcare started. I can't remember all the ins and outs now but I went to a Continuing Healthcare(CHC) training course and basically it was acknowledged that the burden of providing health and nursing care in the community was disproportionately being put onto councils, thus CHC was implemented so that burden is shared by the NHS. If someone is assessed and found needing a certain level of help then they will be entitled to funding from the NHS via CHC.
The problem is that a lot of the problems elderly people incur are more what we call social problems. Many are 'medically fit', but that doesn't necessarily mean at their at full original health and able. This is something that people get a bit confused with sometimes - understandably. Many people ask why we are discharging their parents from hospital because 'they're not well'. The thing is these people aren't acutely ill and their condition isn't going to change, the situation they're now in is their new reality and something that isn't going to improve so they need help adjusting to their new self concept but also have a tailored plan of care in the community that enables them to fulfil their activities of daily living - with assistance or not.
Now in many cases this is for instance, a mobility issue that might make washing and dressing difficult. So they need carers to help them to do activities. They're not ill necessarily. This is social care and thus self funded and/or funded by the local authority.
Now in some cases this is more than that, the person might have a few morbidities and need regular nursing care. For example they might have end stage cancer or exacerbated COPD etc. They may not be suitable for a hospital bed anymore, or may wish to go home off their own back. they'll be assessed to see if the care and input they need is beyond the responsibility of the local authority and if it is, the NHS fund through Continuing Healthcare.
That's a really simplified explanation that touch doesn't upon those with high needs from disability etc but explains it a little bit. Elderly people aren't just left without healthcare. It's just distinguishing what are social issues and what are medical issues, as well as seeing who's responsibility it is to pay.
More about the CHC researchbriefings.parliament.uk/ResearchBriefing/Summary/SN04643