How can the UK honestly view their healthcare as better than that of the US?

I worked in the NHS for about 7 years, so I have some insight that I'll share here.

1. You don't need to hold health insurance in any form.
2. You don't need to give details of your health insurance to be seen by a doctor.
3. You may be the richest or the poorest person, but if you have a heart attack or a stroke or a bad car accident, you will only be treated in an NHS hospital. Almost no private medical facility is equipped to deal with such emergencies.
4. You will always be treated courteously and with consideration no matter your background.
5. You get the same great healthcare whether you pay a million in taxes per year or if you've never earned a penny in your entire life.
6. There is no such thing as ‘pre-existing’ diseases not being covered.
7. There is no co-pay. You don't need to pay back a fixed percentage or an excess no matter how much your treatment cost. If only car insurance were the same…
8. You will almost never be chased by whichever company to pay money back to a hospital or insurance company.
9. You will never become bankrupt because of steep medical bills.
10. You will never have to sell your home to cover your medical bills.
11. You will never fall behind on house mortgage or rent payments because you diverted money towards medical expenses.
12. A doctor will almost never see whether you're eligible to be treated before he or she examines you.
13. By and large, the treatment you get in one hospital is much the same as you'd get at the one in the neighbouring town or city.
14. If you're seriously ill, or unconscious, you will never be dumped at the doorstep of a hospital by family or friend just because they know you don't have medical insurance and neither they nor you can pay for your care.
15. You will never have to submit bills and letters to insurances companies asking them for reimbursement of your medical expenses.
16. Your need to undergo accepted medical practice can never be turned down by some cost cutting insurance firm.
17. An ambulance ride to hospital is completely free. You do now pay a £50 charge if it wasn't needed ie a women in labour can't unless there is an immediate risk to hers or the babies health. 10 years ago we Could phone for an ambulance for anything. Ie a broken arm now they tell you to make your own way.
18. If you are unable to use public transport either due to handicap or poverty, you will get a free ride to your hospital appointment.
19. You'll never fall out of medical cover just because you forgot to mail your cheque to your insurance company.
20. You'll never fall out of medical cover just because you lost your job and your employer-paid insurance.
21. You'll never fall out of medical cover just because you think, “What the hell, I'm perfectly healthy, why should I buy medical insurance?”
22. You pay £8.60 per out patient medicine prescription for up to a month. That's about 12 US dollars. In England this is free if you are a child, unemployed, or a pensioner. In Scotland and Wales this is free for everybody.
23. If you are on certain benefits (tax credits / social welfare) whether unemployed or in low paid employment you also get exemption certificates, which not only make prescriptions free but also include free dental and also free eye testing and a fixed percentage off glasses should you need them. (Thank you for this info Aaron Oxlade).
24. Just because you live close to another country with cheaper medicines, you will never travel every month or two across the border just to buy those cheaper medicines. The only cheap border hopping shopping that the British do is a booze cruise to France.
25. Certain chronic diseases, including cancer, entitle you to free prescriptions, so you live your life free of the emotional and financial burden that comes with having to spend so much of your income on your own health, money that you might otherwise deny your children or parents or partner.
26. Your dependents (children, partner, elderly parents) will never fall out of medical cover because you didn't buy insurance.
27. Just because you didn't pay medical insurance your entire working life, you won't be quoted annual medical insurance of tens of thousands of dollars now that you're old and infirm with hypertension and diabetes.

I could probably go on if I took some time to think about it. This reply has only taken me about 20 minutes to type out on my phone.

To close, the medical system in the US is rather pathetic in comparison to the NHS. Whilst America may espouse the ideal of “all men are created equal”, we Brits, at least in the field of health, actually practice it without a moment's thought.

Please do upvote this answer if you support the NHS!

Support for the NHS is strong among Scots, seen here demonstrating before their parliament.

Update 4th Nov '18: This is my first answer ever to get to 1k upvotes! Thanks everyone, for taking the time to read it all.

CODA: The NHS, like most great institutions of this kind in Europe and the world, is multi-cultural and multi—ethnic.