I think the main problem is that you have the sequence reversed for the most part.  Wealth and capital is first accumulated through capitalism, and only then are the economic conditions created where children no longer have to work, governments can pay for schools, and perhaps even health care.  You can make a law outlawing child labor all you want (every country has one), but unless you have the economic conditions to support it, it is totally meaningless.  Economics dictates reality, not government declarations.  If the choice is between children working or starving, they will work regardless of what the law says.

Under your definition of "socialism," practically every country on the planet qualifies as "socialist."  And yet, non-coincidentally, you chose to identify the wealthy Western capitalist countries as your examples of the success of "socialism."   Curious.  

Strange how "socialism" tends to work best in wealthy capitalist countries eh?