The issue is undoubtedly multifactorial.
I agree with you that we cannot persist with above-inflation spending increases. This is economic fact. Moreover, we're currently enjoying an artificially cheap healthcare system by underpaying nurses and doctors.
My partner and many friends and family work for the NHS, and these are my general observations, in order of significance.
1. The biggest problem is underfunded social care. If we took money out of the NHS and moved it into social care, we'd have a better NHS. At the moment, the NHS is full of inappropriate patients with nowhere to go. We also need to consider how we handle the elderly in this country. The idea of having elderly people stay at their suburban homes while the state pays for nurses and doctors to visit is hugely expensive. I suspect caring for your elderly relatives at home is going to need to become more common.
2. The NHS is inefficient BECAUSE there are to too few admin staff. In NHS terms, this includes IT and other similar roles. Currently, my partner spends over an hour a day just waiting to log in on one of her department's three computers. She then spends ages sending emails, following up transfers and doing a whole load of other jobs that should be done by an administrative person. Technology is massively underexploited because investing in technology costs money that we don't want to spend. It's like that death spiral that I'm sure many of us have seen in our careers, where you get nothing done because you spend all day firefighting, and everything eventually grinds to a halt.
3. We need to focus more on preventative care. Right now we largely leave problems until they are much more difficult to treat. This is in large part cultural, and partly due to the firefighting I mentioned before. We have a very unhealthy population (myself included) and more needs to be done to fix that. Other countries have deployed very successful strategies that we've so far failed to apply ourselves.
4. We need to look more at how we handle the end of life process. Sadly, death is an unavoidable reality currently, and we spend a ridiculous amount of resources trying to slightly delay the inevitable. Many of my partner's patients are in their 80s or older, and it seems borderline immoral to be performing such invasive surgeries on extremely unwell elderly people. Interventions continue to get more incredible, but also vastly more labour and resource intensive. Many of these patients may live an extra few months due to these interventions, but we really need to be honest with ourselves and ask whether it's worth it. Currently an intervention is deemed value for money depending on the average QALY added across all patients, but obviously in reality, the same intervention would be usually much more value for money on a healthy 40 year old, and much less on a frail 90 year old. This is an extremely controversial and morally difficult area with extremely challenging issues around every corner, however.
This isn't a problem of funding, either. This is a problem of basic population dynamics. We could throw hundreds of billions more at the problem, but where are we going to get all these new carers, nurses and doctors? Who's going to be left to build their houses and grow their food? If we import all these people from overseas, which is politically unpopular, how are we going to integrate them?
We could look into whether we continue with the Beveridge Model. Certainly some things about it don't work brilliantly. But honestly I don't think the model matters so much as the fundamentals, other than better aligning incentives will make it marginally easier to fix.