Indeed. If what Trump is quoted above as saying about Americans paying more than us is true, then American buyers should be asking why that is and refusing to pay more than the NHS do.
The thing is, the "buyers" in the US system are the insurance companies.
They choose how much of a deductible (excess) needs to be paid by the patient, and then the rest of the cost is pooled between everyone's premiums. Nothing's really coming out of their pocket, provided that they set their premiums high enough (or choose who to offer insurance to) so that they end up with a profit for their shareholders.
You might think that the ability to offer lower premiums than their competition would be good for business for insurers. But with many individuals getting healthcare as part of their employment package, they don't really shop around. And the large companies buying healthcare policies for their workforce are prone to accepting kickbacks.
Hospitals and healthcare providers aren't funded centrally so their source of revenue is the invoices they issue to their patients and their insurance companies. So it's in their interest to invoice for everything they can. And the manufacturers of pharmaceuticals, equipment, etc, will charge as high a price as they can, especially for any product which is protected by patent, licensing, or some other exclusivity. Add on a whole load of lobbying and bureaucracy, by those that are currently making money from the system, and want to keep it that way. (e.g. manufacturers like to be able to advertise their prescription drugs directly to patients, so lobby to keep the right to do so.)
The trouble is, when you get down to the individual, refusing to pay (for insurance) just leaves you with the danger of a huge medical bill that you have to pay for yourself. And that's not an option anymore, you have to buy healthcare. The recent reforms to US healthcare (
Affordable Care Act of 2010) aimed to make insurers offer healthcare premiums to all, and also required everyone to obtain insurance. But with all the vested interests involved (see lobbying), it's not been a panacea by any stretch of the imagination. And the current US president is quite adamant that the Act needs to be revoked.
(Note, the above is an outsider's understanding of the situation. I can't claim any personal experience of the system, and I wouldn't particularly want to either.)