No, in the United States they're generally passed on to the insured's employer. There are several negative aspects of that, but the big one is that the patient is less likely to complain about being overcharged if someone else is paying for it.
The whole idea of employer-paid insurance in the U.S. is an artifact of some tax law jiggery pokery back in (if I'm remembering right) the Depression/WWII era. Before that people paid for their own health insurance.
Perhaps that's how the mythical homo economicus would see it, but that's not how real people see it.
See also: the way that most people view a tax refund as "free money" that the government is "giving" them, rather than them having given the government an interest-free loan of that money (which is the actual case).
When they get a (partial) refund of the taxes withheld, they're happy, rather than being disappointed that they've had too much withheld (as homo economicus would).
If, instead, no taxes were withheld and they were forced to come up with the (non-refunded) portion in a lump sum payment once a year they'd be outraged.
Human psychology and math don't really overlap that much.
In one sense, yes, the money the employer pays would be available for wages, but the problem is that any savings would come out of a pool of potential wages for all employees. There is no direct incentive for an individual employee to save money on health care, because those savings will make zero impact on his paycheck.
One way of trying to get around this is, of course, the "copay". If the employee saves money, he does, in most cases, have a somewhat smaller copay. But the incentive isn't nearly as high as it would be if all the savings accrued directly to the employee.
The whole idea of employer-paid insurance in the U.S. is an artifact of some tax law jiggery pokery back in (if I'm remembering right) the Depression/WWII era. Before that people paid for their own health insurance.