what does a two-tiered system look like? I keep thinking about private and public schools....
You’re right, more or less private and public.
Many European countries have two tiered system, in the UK for example, people with means do not always use the NHS and may choose a private doctor, for some issues.
From wiki:
“Private health care has continued parallel to the NHS, paid for largely by private insurance: it is used by about 8% of the population, generally as an add-on to NHS services and mostly obtained through employer funded insurance schemes. This is a taxable benefit to the employee (i.e. the value is imputed by the tax authorities as income to the beneficiary). Because NHS services are so comprehensive, there are many areas where the private sector usually does not compete and private insurers almost always refuse to fund. Childbirth and perinatal services are good examples. Conversely, there are some areas where the NHS does not offer free treatment (cosmetic surgery for vanity purposes for example) and thus the private sector offers a pay-for-service alternative.”