The reason for this is that, no matter how loosely-held it is, an electron in an atom has a lower free energy than a free electron. The wonder is, not that neutral sodium is stable, but that it takes so little energy to knock an electron loose. It is smaller than when chlorine gains an electron. However chlorine is actually more stable when it gains an electron. The electron affinity energy stabilizes the chlorine ion.
Electron affinities follow the same pattern.
Actually, chlorine is not quite stable as a neutral atom . Yes, you are. What you are missing is the real and essential interactions between atoms which make up chemistry. To understand a process one must look at all aspects of the process. More about Michael Polanyi's epistemology can be found here .