Yes, the UK needs to pay more and it's needed to pay more for decades. That's precisely the issue. But turning everything into NHS vs. USA has obscured that debate for a long time. The NHS does not compare favourably for outcomes for many cancers.
I'd never advocate American style fees; as well as the cost the way they are done is very confusing. The UK has a higher proportion of healthcare costs paid via public sector than almost any other country, but the amounts (known as cost sharing) vary quite widely between countries. For example someone mentioned that in France a visit to a GP costs about 25€. That's what the health insurance allows as a normal fee. They pay 70% of that. You can pay the rest out of pocket, or you can take out supplementary insurance. If your income is below a certain amount, you receive what's called the CMU, which pays 100% of the normal fee. Some doctors choose to charge more than the scheduled fee, and you pay the gap between the normal fee and theirs--or some supplemental plans may pay it. For a whole host of chronic conditions, public insurance pays 70% (the French system is quite complicated).
Insurance doesn't necessarily mean a system like BUPA or the old American system (nowadays you can't be rejected for health reasons). In some countries everyone has to purchase insurance; it is subsidized and regulated, and everyone is accepted. The Netherlands does this, and Switzerland.
Meanwhile, Canadians are always shocked that in Britain you can jump the queue by paying, which is very limited in Canada (and to do it you have to give up being able to bill the province: no dual practices). But their system does not cover prescriptions or various things like outpatient physiotherapy.