> Single payer in the US simply means the patient doesn't front any money.
No, that's not what it means.
> The doctor/hospital/... just gets the money for medical services rendered directly from insurance (and insurance collects things like a deductible from the patient, not the doctor. Essentially, insurance guarantees payment to the medical provider and deals with all patient financial matters).
You seem to be describing medical insurance in general, not single payer health care.
> Medicare and medicaid are single payer.
No, they're not. "Medicare for All" would be single payer. https://en.wikipedia.org/wiki/Single-payer_healthcare