Here in Oz we have Medicare (the government UHC) and private coverage. The mix of services and payment is a curious thing.
You are guaranteed a base level of coverage. No matter what your income or circumstances, the idea is that you will get care. Hospital says are in state run hospitals, you get a public ward, full of beds with other patients. You do not get to choose who treats you. For less urgent (or elective) surgery, you join the queue.
Private care, you get to stay in private hospitals, usually with a single occupancy room. You choose your doctor. Access to less than urgent treatment is vastly faster. Private cover includes ancillary services - physiotherapists, and the like. There is however often a co-payment for these. Private cover may be extended to include dental, although the co-payment is of the order of 40%, or even more for crowns and bridges. Government dental cover does not exist, and remains a political football. (You can, and many do, attend a clinic at the dental school, which is free. You are of course seen by students under supervision.)
Now the weird one. No matter whether you are privately insured or not, your doctor is paid, at least in part, by Medicare. There is a government ordained scheduled fee, and Medicare will pay betweem 100 and 75% of that. So even with UHC, there is a co-payment. However a doctor may elect to not bill you, but bill Medicare (termed <i>bulk billing</i>). In which case he gets the 75/85%, and eats the gap. Many GPs bulk bill all their patients, especially in less affluent areas, and pretty much always where there is any hardship. (One win for them is that they are assured of payment from Medicare, and don’t have to go chasing bad debts or late payments.) Doctors are of course free to charge more than the scheduled fee as well. In which case you get to pay the gap. But you can shop around. If you want treatment from a specialist, you can’t get a rebate from Medicare unless you have first seen a GP and got a letter of referral.
It is illegal to insure for the gap. Unless the treatment is whilst admitted to hospital. So for hospital stays, you can get full personalised treatment from a doctor of your choice in a private room in a private hospital (and some look more like five star hotels than hospitals) with little to no waiting, even for non-urgent treatment, essentially at no fee. If you have private cover.
It is accepted that for some things, you will be in the public system no matter what. A serious motor accident will have you in the nearest big public hospital’s ER.
In Oz, prescription drugs are already seriously subsidised by the government. For drugs that are approved we pay a capped prescription cost of about $30. If you go over $1000 of prescribed drugs in a year, the cost drops to $5 per script. The private insurers do extend coverage for some drugs not on the schedule, but since the coverage is actually pretty good, this isn’t an issue for most people.
The Oz government really likes people to have private coverage. So much so, there is a 30% rebate on the cost, which can be claimed against your income tax. The government breaks out some of the costs for Medicare as a separate component of your income tax (it isn’t true really, the real cost to the taxpayer is higher) and if you don’t have private cover, they increase this amount.