... nowhere in the definition of a free market can you see that there is some guarantee that everyone can achieve survival ...
Nowhere in the definition of a
government can you see that there is some guarantee that everyone can achieve survival, and in fact governments do rather poorly at this.
The very poorest segment of society (such as those who sleep rough on the streets) accesses far less of the state's resources than the other segments. Many government programs need you to have an address, or need you to have a bank account, or need you to have a computer, or need you to have a birth certificate, or need you to have ready access to transport, or need you to be literate.
Statist laws take away much of the ability for the destitute to improve their life. In the UK, many old people are very poor, many are really struggling to stay warm and get enough to eat. And yet old people can be the most valuable child carers. They have plenty of time and patience, flexible availability, and lots of wisdom to pass on. But current laws make it almost impossible for them to care for children on a casual basis. Even if they can meet all the paperwork requirements and the bureaucratic inspections, it can take many months for the permission to come through.
Younger people suffer too. In past centuries, a youth who has fallen on hard times could improve his lot through hard work, starting one step at a time with "odd jobs" like chopping firewood. But employment is so regulated nowadays that it's not worth anyone's while to pay people for odd jobs. For the unemployed, it's "all or nothing", whereas humans are best at getting themselves out of a bad situation if they can improve their life incrementally.
The number of people who are truly unable to survive without welfare is very small, and those people do better under voluntary charity than under state bureaucracy.