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.
> 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
Every wealthy country besides Canada that provides public care is a counterpoint to this.
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.