There is also no successful example of a purely capitalist state with no elements of socialism.
If by "no elements of socialism" you mean no social support structure, then yeah. But if you mean no state-operated social programs, whoo, buddy are you dead wrong.
The USA has always been socialist. FDR was a socialist and socialism is necessary when a country reaches a certain size. I think the population size and size of a nation necessitates socialism due to the mere fact that people who don't know you wont care about you unless forced to and while small towns where everyone knows each other don't need social programs, large towns where no one knows each other or where one part of the country doesn't know the other part, this requires socialism. It's just a matter of the kind it requires, authoritarian sociaiism is not the kind of socialism I endorse.
But there should be a basic income. That should be mandatory both for economic reasons (it's better for the economy if everyone has at least some money to spend), and humanitarian/human rights reasons. Human rights are important and the basic income helps to protect the human rights of those who receive it.
I think also education or the opportunity to self improve has to be considered a human right. If it requires socialism to have that then so be it, but if we don't allow people to work hard and improve themselves over time then we are wasting their lives, skills and talents and it will show in our economy. Already the USA suffers from ignorance due to the fact that education is so expensive.