The current system does exclude people. It mostly doesn't do it by fees, it does it by queuing. Example: In my part of London, depending on the time of year, an appointment with the GP can take between 2-3 weeks. There have been several occasions when I've wanted to see the GP to get some issue checked out/get reassurance that it's not serious/etc., but have given up because 3 weeks was long enough for the symptons to go away (you could argue that if they go away in 3 weeks then it probably wasn't serious, but that's still 2 weeks or so of the worry of not knowing). Another example is the huge waiting lists for operations (although I don't have figures, I'd be astonished if there aren't some people who die while waiting months and months for an operation).
Similarly, the stories we are hearing - some posted on this thread - of the situation in many A&E departments is appalling - and it would again be astonishing if at least some people who ideally ought to get A&E treatment, or ought to be checked out after a minor accident, get put off from going to the local hospital because they know of the pressure A&E departments are under/can't wait the 4 hours or so. (Obviously this wouldn't be the case for people who are very seriously hurt, I'm talking people who have been mildly hurt - enough that getting a doctor to check them out would be a good idea. It's also more likely to happen with people in the kind of low-paid jobs or zero-hours contract where waiting hours on end in an A&E department will mean you'll be late for work with a risk that your employer will just not to give you any more shifts if you are late).
Ultimately, whether you like it or not, resources for treating people are finite, and it's simply not possible to treat everyone for everything for free. There has to be some limit. You can push that limit back by pumping more money into the NHS, getting more doctors and nurses, etc. You could do a lot by funding social care properly - and I do think the current Government's record on that is truly appalling. But you can't change the fact that that limit is ultimately going to be there at some point. There is no way to avoid the problem that, no matter how well you fund the health service, sooner or later you'll hit a financial boundary at which you'll need to prioritize what treatments are available, or whether some treatments that are perhaps non-essential can be funded by other means. And I don't think there's anything wrong with acknowledging that, and having a discussion about where that boundary should be or what the priorities should be.
In that context, it's clear that the NHS is now being required to spend huge amounts of money treating conditions that would have been entirely preventable with - for example - better lifestyle choices. I think there is a legitimate argument about cases where people require treatment for things that are arguably largely self-inflicted, or where the person shows no sign that they are prepared to modify their lifestyle to prevent the problem recurring - although I realize that the boundary there is very fuzzy and it's very hard to see where you can draw the line in a way that doesn't look unfair to some people. I also wouldn't go along with the judgemental manner in which some posters here have been presenting that argument.