OP I think it's a stretch to suggest that the NHS is anything like the very basic care provided for non payers (the poor) in the US healthcare system!
Your anecdotal experience isn't the picture across the country. In my part of London it's annoying having to go online at 8 am to book a same day GP appointment, when the alternative is calling up listening to the countdown for 15 minutes to be given an appointment in three weeks.
The hellish NHS keep reminding me of my smear test by text and my ageing parents of their annual flu jabs. The feckless GP who has little to do and whom I really don't see much because its a massive over-prescribed practice has the audacity to telephone me out of the blue to prompt be to have my smear!
I've visited hospitals that seem run down and that are smelly dark places but also ones that are clean, bright and where staff don't look close to blowing their top through long hard hours.
That's just London. I have family in Colchester, where they think I'm long-suffering!
But then I'm not too ill. I avoid a&e because it's for really sick people.
Sure, healthcare services shouldn't differ town by town, region by region but they do. There should be more local NHS GP practices and healthcare centres specialising in mother & baby, elderly, sexual health, and blood test centres. It shouldn't all rest on Hospitals. Not with the numbers we have in our major cities.