Listener:
Is a drunk person putting others at risk if they are not at home and not driving?What about a mentally challenged person, or a drug taker, whether prescribed or not. When they pose a serious risk because of their proven social misbehaviour something can be done about it but usually not until they've already caused damage.
What if the unvaccinated person is prepared to take extra precautions so as not to be any more of a risk than a vaccinated person and what exactly would be required? Who are they a risk to, another unvaccinated person or a vaccinated person?
It isn't really the purpose of analogies to cover every possible scenario. Nor does something being analogous automatically mean that both things are equally valid. It's just a metaphor for quickly comparing ideas. Most readers with a fairly basic education would understand the intent of the comparison the doctor was making.