I think it is possible that the lack of insurance has no effect on aggregate mortality statistics. I do not think that this is likely, but I think it's possible. What I think is likely is that the effect is not that large, because if it were large, it would be very surprising to see so little effect on the mortality of an elderly population with a high mortality rate, or to have a study that samples 600,000 people and finds no effect.
Mostly what I think is that the statistics are really, really flawed. Not because the authors are bad social scientists, but because this stuff is so hard to tease out. Natural experiments are rare, and data sets often hard to come by.
I'm much more convinced by the benefits of health insurance for certain subpopulations, particularly people with diseases we're very good at treating. HIV seems to pretty convincingly respond to offering public treatment--which also has a pretty compelling public health rationale. (I don't want to hear anything about spears mounted on steering wheels, thank you very much). Medicaid expansions provide some pretty good natural experiments, IMHO, indicating that you can improve infant mortality. Poor people with hypertension get better blood pressure control pretty consistently.
But this doesn't imply a large effect in the macro data if we extended health coverage, precisely because not that many people under the age of 65 die of things we can treat. That whole age group is only about a quarter of deaths, and some of them are from things like metastatic cancer or auto deaths, in which more health care coverage can at best eke out moderate further improvements. (That may not be true in the future. It is now, sadly.)
Obviously, this matters. If 45,000 people die a year, this makes a more urgent case for overlooking the drawbacks of single payer than if 1,000 people die a year--there are probably more cost-effective ways to control those deaths
But the core question for me is not whether there's any effect--I'm willing to consider the possibility there isn't, but I tend to assume there is. The question is, how big? Because if it exists, but it's too small to measure, it might not be the issue our government should be most focused on. Particularly when you consider that there are costs, as well as benefits, to a national health care system.