A simple way around that would be to pay for them so that no patient was billed for one.

Is there not a problem that in the US it would encourage people who otherwise couldn't afford the operation to opt for the "riskier" trial, even if they'd really prefer not to. Whereas in the UK, they would have received the operation free in or out of the trial, therefore the patients financial situation wouldn't factor in to their decision to join. Which feels significantly more ethical to me.

In theory you either have insurance that should cover this (more than 90% of Americans are covered), or else you'd be too poor to really even be able to afford such an operation or medical care. The system sucks, but I don't think the way the system is set up would cause a significant number of people to use free research for medical alternatives.

Even if you have insurance you get a statement from your insurance provider detailing how much everything cost and how much of it insurance paid for.

People in a clinical trial typically don't pay for the treatment (or placebo).