Just read the worries expressed on this forum about the precarious state of access to health insurance in ER and it is easy to understand why many of us believe universal access is an imperative.
I think the discussion is more valuable if we avoid imprecise or misleading terms. As an example: "universal access" is precisely what we have today. Anybody can have access to any medical care. Sometimes people use the term to mean "everybody gets medical care that they don't have to pay for as an individual service."
Some proponents of nationalized/socialized/single-payer systems are actually against universal access, as they would prohibit universal ability to buy the health care an individual might desire.