Right now, a member of Congress has to buy health insurance through the DC exchange. (They aren't exempt from Obamacare.) He's also Medicare eligible, and that's probably covering his care. However, he can't really give that to all Americans. He's getting the government to pay for medical care that is generally operating in a private context and mostly financed privately. If he gave everybody Medicare, that dynamic would definitely change at least in many respects.
That's the hard part. However, I think people would be more willing to go along with the costs (meaning much higher taxes) if they had more faith that the government would do a respectable job. When most people think of government, they think of expensive but mediocre schools, shabby infrastructure, long lines at the post office, and long lines at the DMV. They don't want their healthcare to look like that. People do go along with it in Europe because those governments have a better track record.
By "leverage," do you mean trying to negotiate down costs? If so, they definitely do that with Medicare when it comes to providers. That's why reimbursement rates are so low. However, they don't do it with prescription drugs. Republicans don't create entitlement programs often, but when they do, they ensure that it's unfunded and that it gives the taxpayer the crappiest deal possible.
