‘Welfare capitalism’ is just another way of saying democratic socialism. Democratic socialism is an economic system common in Europe and Canada to a certain extent and Australia. Where you have a robust private sector private enterprise economy. But you also have a large publicly funded welfare state that are paid for by high taxes and a lot of taxes (at least by American standards) to fund that welfare state. It is just that people in America who would be called Social Democrats or even Socialists in Europe don’t like using terms like democratic socialism in America because of the negative political stigma that comes with the words socialism and socialist.
That welfare state funds everything from education K-12 or whatever the grades are depending on the country. To college, to health care, to health insurance, unemployment insurance, maternity leave, sick leave, childcare. Things that Americans except for the education part tend to get through the private sector. The United States generally spends around twenty-percent of its Gross National Product on the Federal Government. Europe is generally around fifty-percent on their federal government depending on the country.
This socialist form of capitalism (and yes there is such a thing) has been something that Social Democrats in America have tried to bring to America since the New Deal era. But haven’t had any real success since the Great Society of the 1960s. Because for one the country since the late 1960s if not further back has moved right politically. But also Americans tend not to be as far to the left as Socialists and don’t like the idea of paying a lot in taxes to pay for services they can get in the private sector.