Here's one good reason: Europe. Most of Europe has 'free' healthcare. But they found out very quickly that in fact, just taxing the rich wasn't enough to pay for it. So they implemented a VAT. Almost every country that has socialized healthcare has a VAT. And a VAT is a HIGHLY regressive tax, falling mostly on the poor and middle class. But it's also the one tax that is efficient enough and sneaky enough to raise enough money to pay for govt healthcare. Have you ever gone to Europe and wondered why everything is so expensive? The VAT is why.
What's more, 'free' healthcare systems are inferior. Oh, I know every socialist tells you this isn't true. They probably site infant mortality and life expectancy statistics to prove it. But have you ever wondered why they cherry pick THOSE stats? If you think about it, they're actually NOT very good proxies for healthcare quality.
Every developed country EXCEPT the USA rigs it's infant mortality stat. How do you think that stat is calculated. # infant deaths divided by # infant births, right? In the US, that is how it's calculated. In countries with social medicine though they exclude any baby that is born underweight OR dies too soon after birth. Yet those are the VAST majority of infant deaths. Most babies with health issues come out underweight and die quite quickly. Studies that do apples to apples comparisons find that the US healthcare system is FAR AND AWAY better at preventing infant mortality than social systems.
Life expectancy is also affected by that cheat, but there's an even larger one. Life expectancy is heavily influenced by the US obesity problem. Obesity is strongly correlated with dozens of health problems. Every doctor tells their US patients to lose weight. Is it healthcare's? fault that patients don't listen? Studies that compare health outcomes of people with similar diagnoses find that the US system far exceeds social systems in curing illnesses.
You don't even have to read the studies for this. Have you noticed how often there is a story about some European or Canadian wealthy person coming to the US for medical treatment? Why would that be if the systems were of similar quality or if social systems were better? It wouldn't. How often have you read about the opposite. I can't think of one. Social healthcare systems are great for routine checkups and E.R. care. They SUCK at curing serious illness. They suffer from lack of specialists and lack of specialized equipment that are commonplace in the US.
Now, none of this is to say that the US system doesn't have problems, but socializing solves only one of them, and only at ENORMOUS cost. The one problem it solves is that the US system has a coverage gap between people poor enough to get medicaid and people whose employers provide coverage. ObamaCare was supposed to close that gap, and it narrowed it a bit, but that's about all, and it introduced many serious problems to do it.
The fix is to go the other way. To REDUCE government involvement in healthcare. Go read up on Singapore's health system. It's not a perfect market. Government is involved. But unlike most government involvement, Singapore's healthcare interventions work hard to maintain market style choices and tradeoffs that encourage people to restrain costs and vendors to compete. THAT is what we should shoot for.