there wouldn't be any way to compete with tax pre-paid $0 point of sale health care, so there would be no option. the term "public option" is a weasel word. there would be no private option because nobody is going to choose to pay out of pocket in addition to their taxes that already pay for care.
advocate for whatever but use honest terms. you're advocating for a single payer system and there's no evading that.
Fwiw, a lot of European healthcare has both a public and private option. You may pick the private options because they are "better" in some ways (e.g. more modern clinics, shorter waiting times, or sometimes just better care) which still leaves some wealth gap, but usually means no one goes bankrupt to cure cancer.
Usually the medical part of it all is strictly worse in the private sector (at least in my country) because the public system has a pretty strict competitive exam to get in, whereas profit driven private companies hire the cheapest doctor they can get.
Not everybody realizes that and they often fall for the single room in the hospital.
Shorter waiting times is definitely a thing though, especially for non life threatening conditions.
It's quite country dependent. For example in Hungary my understanding is that many doctors have both private and public practices, but private clinics often don't have expensive machines. Doctors in public hospitals are severely underpaid so they have strong incentives to move you to private practice.
In Italy doctors also have public and private roles but can practice privately in public hospitals, which is weird but was an attempt to avoid losing them to private clinics, for the same reason. You also have private clinics administering public healthcare with a minor markup paid by the patient, and the base rate paid by the state, which isn't a thing in Hungary for example.
It is in general for non-life threatening conditions that there's such competition tho, I agree.
In the UK a large proportion of the doctors are the same. Sometimes even using NHS operating theatres, or with NHS trusts running the clinics, as they are allowed to run for profit services to supplement their budgets...
The UK has both universal healthcare and private options that are far cheaper than the US.
Most universal healthcare systems coexist with private options paid separately. Some are provided by private healthcare providers, and then too tend to coexist with privately paid services.
It's silly to say "having options but you probably won't choose them" is the same as "no option".
There's probably a decent point nearby about subsidized (or tarriffed) work messing with the benefits of free market pressure but it isn't this silly overstatement.
advocate for whatever but use honest terms. you're advocating for a single payer system and there's no evading that.