The songs Button up your overcoat and Take Good Care of Yourself both express the idea that the person addressed has an extra reason for 'taking care' because the singer cares about them.
The use of 'take care', even when it's said to people who are not especially close to the speaker, carries this implication of 'take care of yourself, I want to meet you again'; in other words the speaker is well-disposed towards the person addressed.
The answer to one of the OP's questions is 'no', people do not automatically take care of themselves. They do risky and careless things all the time like crossing busy streets without waiting for the lights, getting drunk, underdressing in cold weather, failing to apply sunblock in sunny weather, taking shortcuts down dark alleys in dodgy parts of town, driving too fast, using mobile phones when driving... The list is endless.
The phrase is a secular equivalent to 'God be with you' which is shortened to 'goodbye' and has, therefore, lost most of its meaning. 'Take care' is also, perhaps, more appropriate to a culture of self-reliance where you are expected to protect yourself. I get the feeling that it's American in origin which fits with that self-reliant idea.