As someone who lives in a socialist country (well, we ALL live in socialist countries to some extend, but a most socialist one than the US), I disagree. The goal ISN'T to "give people every chance to exceed their reach", as you put it, but to make sure people can always pick themselves up and keep contributing to society. To make it so that when life isn't fair it isn't as bad as it could be and it isn't the end. If you lose everything, you still have access to health care, to shelter, to services to help you find a job and work to get it back.
EI exists so that, if you lose your job, you won't find yourself on the street if you can't find a new one before next month's rent it due. Public healthcare is there so that if you're only making minimum wage, and you develop a heart condition, you won't die from a lack of funds to fix it. Education subsidization lets you develop skills to let you better your own life and at the same time better contribute to society as a whole. Socialism is the mantra of "today you, tomorrow me" restructured into a political system. It's an acknowledgement that everyone in a society is in a symbiotic relationship with everyone else in it and that working together is better than killing each other over every last scrap we can personally get our hands on.