The government should be in the healthcare business since the nature of the service precludes a functional market.
For instance, if you believe that we have a right to live, and healthcare has some relation to life, then haelthcare might arguably be a right.
Another might be an analogy, such as with law enforcement. Some might say we have a right to a life free from crime. Most jurisdictions prefer to run and staff their own police departments, or contract with another government-run police department. Only as a last resort would privately-run police departments be acceptable. Additionally, police officers, given their mission and tools, have the power of life and death over the citizens they serve. This is a typical reason given for government-run law enforcement.
Healthcare, to a certain degree, also has the power of life and death. Therefore, one could argue governments should run healthcare.
You can fall into the trap of assuming your "right to private property" means that taxation is theft or you can start with a "right to privacy" and decide law enforcement should never be able to access the contents of your smart phone, rather than discussing the reasons for private property or privacy.
And in case case, rights tend to never be justified, just asserted and treated like true axioms.
I think the definition is a very important distinction, although you're right about outcomes being a binding agent.
I don't believe that health care is a right. A right is something that another person doesn't provide. This "me" movement has really clouded everyone, thinking they are entitled to everything, even if what they want to be entitled to compels another person to provide it, thereby infringing on that person's rights.
So while I don't think it's a right, healthcare is a good idea as an investment and is a social good.
So I agree on the positioning you suggested to get both sides to come together. It's the right thing to do and look at all the tangible benefits we could get from it....
So you're right that talking about rights is unhelpful, I'd go further and suggest that just making healthcare provision better, rather than more/less market-based should be the main discussion to further avoid the conversation bogging down into "reasoning about ideology, rather than outcomes".
That is nonsense. To claim to focus solely on 'outcomes' is to simply make the ideology behind the recommendation implicit and outside the bounds of debate.