Healthcare isn't perfect anywhere in the world whether private or public.
And even the US healthcare system isn't really market driven.
Opening with "let's have a market" isn't opening with a cogent position, since rational actor economic theory is completely unable to deal with this (sure, there might be others, but no one making this assertion wants to talk about them).
My premise is that you can never spend enough money on healthcare. There is always a need for better treatment, more treatment, new machines etc. Even if you taxed people 100% you could still spend more money because as you say it's worth everything to save it.
But a political system is made to prioritize the resources that we have and so a singlepayer system cannot escape to have to come to terms with the boundaries of the budget it's given.
There are many areas where a patient will not be treated or where the system decides it's not worth continuing to pay for some treatment.
In a singlepayer system it gets decided what gets treated in a system which is more open to demand the market decides.
None of these are without fault it's just important to keep their limitations in mind. This is why ex. the french system as much rave reviews it's gotten also sets limits to who it will treat.
We can then have a discussion about whether it's better that everyone gets access to the same treatment or whether it's better that some people can get access to better treatment methods than others.
But to ignore that a single payer system comes with it's own problems such as limits on how many people can be treated and for what and how long, is not taking the discussion serious.