The answer to your question is over 46.6 million Americans..., the ones without health insurance

As the sibling replies noted, everyone in America does have access to at least basic healthcare. If a homeless person is brought into the emergency room after being hit by a car, he won't be denied treatment. The Medicaid program in America reimburses hospitals (and other providers) for charity and bad debt expenses.

Right now 25% of every health care dollar spent is going towards administrative expenses

I've got an unusually close view of this, as my wife is the manager of budget and reimbursement at a largish urban hospital. Most of her job is dealing with Medicaid and Medicare, getting the hospital reimbursed for caring for these people. The reason that she spends most of her (and her department's) time on this is because of the illusion that M&M have efficient administration.

In reality, Medicare and Medicaid account for a huge portion of cases in an urban hospital. They're an 800-lb gorilla: since they account for so much of the hospital's business, the hospitals have little choice but to provide whatever administrative functions that M&M demand. For example, every year my wife must submit the dreaded "Medicare Cost Report". This is literally a crate of documents that my wife's team creates (it takes the team weeks) to document their Medicare charges over the past year. Some time later, the government audits them on this entire crate.

What I'm trying to illustrate is that the actual administrative expenses of Medicare and Medicaid are not really lower than our traditional model. It's just that the regulators force the participants to run it in such a way that the providers bear a large portion of the administrative burden. Since it's on the providers, the costs don't show up on M&M's reports as admin expenses. Instead, they contribute to higher prices billed for the underlying services.

When you look at the big picture, Medicare and Medicaid are certainly no more efficient than conventional providers, and possibly less so.

For example, one of the concerns in rising costs is unnecessary tests. However, M&M regulations tend to cause this. They will refuse to cover the costs associated with various types of infections (e.g., urinary infections) unless the provider can prove that the infection was already present when the patient was admitted. This effectively forces the hospital to test all patients for UTI at the time they're admitted, which is completely unnecessary for actual medical reasons.

Any expectation that total costs would go down due to some miracle of government administration is pure fantasy. Seriously: have you ever been to the DMV, or seen an Immigration office? Why would you believe that a private insurer, given the opportunity to increase profits by lowering their own admin expenses, would not do so? The idea that government-provided healthcare, or government-single-payer, would lower costs really is magic. Just like a belief in the magic of Tinkerbell and Harry Potter.

(Note: I lumped Medicare and Medicaid together for simplicity. While these are related, they're separate entities and anybody who knows the details will probably object. But there's only so much space here.)